i LI BRARY O F CONGRESS, ' 



! UNITED STATES OF AMERICA.* 



.-.-.- ... . . ._ 



NAVIGATION 



AND 



NAUTICAL ASTRONOMY. 



PREPARED FOR THE USE OE THE U. S. NAVAL ACADEMY. 



SECOND EDITION. 



NEW-YORK : 

X>. VAJSC 3STOSXJK^JSTI>, 19£ BROADWAY. 
1865. 









Entered, according to Act of Congress, in the year 1865, by 

D. VAN NOSTRAND, 

in the Clerk's Office of the District Court of the United States, for the Southern District 

of New- York. 



John A. Gray & Green, 

Printers, 

& 18 Jacob Street, New-York. 



NOTICE. 

This Treatise was originally prepared by Prof. Chauvenet to be 
used in manuscript by the students of the Naval Academy. With 
Bowditch's Navigator, oral instruction, the use of instruments, and 
computation of examples, it constituted the course of instruction 
in Navigation and Nautical Astronomy. 

In its printed form, some subjects are more fully discussed, others 
introduced, and various suggestions given on points of practice. 

In this edition examples are supplied, which will serve both as 
illustrations, and as forms for the arrangement of computations. 
Those in Nautical Astronomy are mainly adapted to the Ephemeris 
for 1865. 

It has been my purpose, as I should find time from incessant offi- 
cial duties, to prepare a more complete work, or to supplement it 
with a treatise on the practice of Navigation. 

J. H. C. COFFIN, 
Prof, of Astronomy, Navigation, and Surveying. 

Naval Academy, August 1, 1865. 





GREEK 


LETTERS. 




A a 


Alpha, 


N v 


Nu, 


B /3 


Beta, 


S I 


Xi, 


r r 


Gamma, 


o 


Oniicron, 


A d 


Delta, 


n re- 


Pi, 


E s 


Epsilon, 


p P 


Eho, 


z c 


Zeta, 


S a 


Sigma, 


H v 


Eta, 


T T 


Tau, 


e e 


Theta, 


r u 


Upsilon, 


i t 


Iota, 


<p 


Phi, 


K x 


Kappa, 


X X 


CM, 


A X 


Lambda, 


¥ ip 


Psi, 


M ix 


Mu, 


Q co 


Omega. 



COHTTETsTTS. 



CHAP. 

I. The Sailings : Plane, p. 7 ; Traverse, p. 11 ; Parallel, p. 13 ; Middle 
Latitude, p. 14 ; Mercator's, p. 21 ; Correction of the Middle Lati- 
tude, p. 32 ; Mercator Chart, p. 34 ; Great-Circle Sailing, p. 38 ; 
Shaping the Course, p. 49. 

II. Refraction, p. 51 ; Radius of Curvature of the Path of a Ray of 
Light in the Atmosphere, p. 58. 
Dip of the Horizon, p. 60 ; Distance of the Horizon, p. 63. 
Parallax, p. 65. Apparent Semi-Diameters, p. 68 : Augmentation 
of the Moon's Semi-Diameter, p. 71. 

III. Time : Sidereal, Solar, Apparent, Mean, p. 72 ; Equation of Time, 

p. 74 ; Astronomical, Civil, and Sea Time, p. 74 ; The Relation of 
Local Times and Longitudes, p. 76. 

IV. The Nautical Almanac, p. 80 ; Interpolation, p. 81 ; To find a re- 

quired Quantity for a given Mean Time, p. 84 ; Sun's Right Ascen- 
sion, &c, for Apparent Time, p. 86 ; Mean Time of the Moon's 
Transit, p. 91 ; Of a Planet's Transit, p. 93 ; Right Ascension and 
Declination of the Moon or a Planet at the Time of Transit, p. 94 ; 
Greenwich Mean Time of a Lunar Distance, p. 94. 

V. Conversion of Time : Mean and Apparent, p. 98 ; Mean and Side- 
real Time Intervals, p. 99 ; Mean and Sidereal Times, p. 100. 
Relation of Time and Hour-Angles, p. 107. 

VI. The Astronomical Triangle, p. 114. 

Altitude and Azimuth for a Given Time, p. 117 ; Azimuth from an 

Observed Altitude, p. 120 ; Amplitude, p. 122. 
Hour-Angle and Local Time from an Observed Altitude, p. 127; 

Hour- Angle on, or nearest to, the Prime Vertical, p. 135. 

VII. Latitude from a Meridian Altitude, p. 138; An Altitude at any Time 

p. 144 ; An Altitude near the Meridian, p. 149 ; (Note p. 253 ;) 
Circum-Meridian Altitudes, p. 154 ; An Altitude of Polaris, p. 161. 






11 CONTENTS. 

CHAP. 

VIII. Chronometers, p. 165 ; To find the Rate of a Chronometer, p. 167 ; 
Comparison of Watches and Chronometers, p. 168 ; To find the 
Chronometer Correction from Single Altitudes, p. 170 ; Double 
Altitudes, p. 171 ; Equal Altitudes of a Star, p. 176 ; Equal Alti- 
tudes of the Sun, p. 177 ; Equal Altitudes of the Moon, or a Planet, 
p. 181 ; By Meridian Transits, p. 186. 
Longitude, p. 187 ; By a Portable Chronometer, p. 188 ; Reports of 
Longitudes, p. 190 ; Finding the Local Time from Single Altitudes, 
p. 191 ; Double Altitudes, p. 194 ; Two Altitudes of the Sun near 
Noon (Littrow's Method), p. 198 ; Equal Altitudes, p. 203 ; From 
Meridian Transits, p. 207 ; Longitude from Lunar Distances, p. 208 ; 
Reduction of an Observed Lunar Distance, p. 209 ; Other Lunar 
Methods, p. 218. 

IX. Finding a Line of Position (Sumner's Method), p. 220. 

Latitude and Longitude by Two Lines of Position, p. 224 ; Reducing 
an Altitude for Change of Position, p. 234 ; Latitude and Longitude 
from Two -Altitudes, the Declination being the same, p. 236 ; 
Douwes's Method, p. 246 ; Two Altitudes near the Meridian, p. 251 ; 
Two Altitudes near the Prime Vertical, p. 255 ; Two Altitudes, the 
Declination being different, p. 257 ; Two equal Altitudes, p. 268 ; 
Two Altitudes, when the Distance of the Objects is measured, 
p. 268 ; Three Altitudes of the Sun, p. 269. 

X. Azimuth oe a Terrestrial Object, p. 275 ; Finding a Meridian Line, 
p. 275 ; Azimuth of a Terrestrial Object, by a Theodolite or other 
Azimuth Instrument, p. 278 ; By a Sextant (sometimes called an 
Astronomical Bearing), p. 279. 



NAVIGATION 



CHAPTER I. 

THE SAILINGS. 

PLANE SAILING. 

1. Suppose the compass-needle constantly to point to the 
north, a ship which is steered by it upon any given course 
must cross every meridian at the same angle, namely, the 
angle given by the compass. She does not sail on a great 
circle, except when she sails on the equator, east or west, or 
on a meridian, north or south. All other great circles inter- 
sect successive meridians at varying angles. 

A line which makes the same angle with each successive 
meridian is called a loxodromic curve; in old nautical works, 
a rhumb-line y more commonly, the ship's track. 

The constant angle which it makes with the meridian is 
the course, and is called the true course, to distinguish it 
from the compass course. 

The length of the line considered, or the distance sailed, 
is called the distance. 

The corresponding increase or decrease of latitude is the 
difference of latitude. 

The distance between the meridian left, and that arrived 
at, measured on a parallel of latitude, is the departure on 
that parallel. 

The distance between these meridians, measured on the 
equator, is the corresponding difference of longitude. 



8 



NAVIGATION. 



2. The following notation will be employed ; the refer- 
ences being to Fig. 1, in which C A represents a portion of 
a loxodromic curve : p 

C=BCA, the course. 
cl=CA, the distance. 
I = C B = E A, the difference of 
latitude. 

p = the departure : C E in the 
latitude of C, B A in the 
latitude of B, F G in the 
latitude of F. 

D = C A', the difference of longi- 
tude. 

L = C C, the latitude left, ) + w; 

Ij'= A' A, the latitude arrived at, j — w 




: A' A, the latitude arrived at, 
X =z the longitude left, 
A'= the longitude arrived at, 

Evidently l = Z r —Z, 
whence Z/'= L + /, 



hen North, 
when South. 



\- 



when West. 
when East. 



k'—X +D, 



(1) 



in which attention must be paid to the signs, or names.* 
These formulas accord with the precepts on page 50 of 
Bowditch's Navigator. 

3. If the distance is very small, so that the curvature of 
the earth may be neglected, then C A may be regarded as a 
right line, and the triangle C A B as a right plane triangle. 
From this we have 



cos C r = 



Bin 






tan <7='4f; 



I 



or, 



I =z d cos (7, p = d sin C, p = I tan (7, 



(2) 
(3) 



* If N. and W. are regarded as positive, S. and E. are negative, and may 
be treated as such, without the formality of substituting the signs 4- and — . 



i 




PLANE SAILING-. 9 

in which p is the departure in the latitude of C or A ; indif- 
ferently, as their distance is very small. 

The Traverse Table, or Table of Right Triangles, contains 
I and p for different values of C and d. Table I. in Bow- 
ditch's Navigator contains I and p for each unit of d from 1 
to 300, and for each quarter-point of C. 
Table II. contains them for each unit of d 
and each degree of C. 

These quantities form a plane right tri- 
angle (Fig. 2), in which 
d is the hypothenuse, 
C oiie of the angles, 

I the side adjacent ) _ 

,, .; 9> that an^le. 

p « opposite) S F . g2 

In the Tables, the columns of distance, difference of lati- 
tude, and departure, might be appropriately headed, respect- 
ively, hypothenuse, side adjacent, and side opposite. 

4. The first two of equations (3) afford the solution of the 
most common elementary problem of navigation and survey- 
ing, viz. : 

Problem 1. Given the course and distance, to find the 
difference of latitude and departure, the distance being so 
small that the curvature of the earth may be neglected. 

These equations also afford solutions of all the cases of 
Plane Sailing. (Bowd., pp. 52-58.)* 

5. Problem 2. Given the course and distance, to find the 
difference of latitude and departure, when the distance is so 
great that the curvature of the earth cannot be neglected. 

Solution. Let the distance C A (Fig. 1) be divided into 
parts, each so small that the curvature of the earth may be 
neglected in computing its corresponding difference of lati- 
tude and departure. 

* The first and sixth are the most important. 



10 NAVIGATION. 

For each such small distance, as c a, 

l — d cos (7, p — d sin C. 

Representing the several partial distances by df 15 d^ c? 3 , 
&c, the corresponding values of I and p by ? 19 l 2 , Z 3 , &c, and 
jt? 1? jp 2 , j9 3 , &c, and the sums respectively by [cZ], [7], [j?], we 
have 

?i+4+ ?3 + &C. == (^H-c^ + c^ &c.) cos (7, 
J Pi+i>a+#i + &c.= (d^ + dk + dk &c.) sin (7/ 
or, 

[l]= : [rf] cos (7, 

l>] = [*] nn a 

Since the distance between two parallels of latifliae is the 
same on all meridians, the sum of the several partial differ- 
ences of latitude will be the whole difference of latitude ; As 
in Fig. 1. 

CB = EA=: the sum of all the sides, c 5, of the 
small triangles ; 
and we shall have generally, as in Prob. 1, whatever the 
distance, <#, 

l — d cos (7. 
We also have 

p == d sin (7, 

if we regard p as the sum of the partial departures, each 
being taken in the latitude of its triangle ; so that the differ- 
ence of latitude and departure are calculated by the same 
formulas, when the curvature of the earth is taken into ac- 
count, as when the distance is so small that the curvature 
may be disregarded ; or, in other words, as if the earth were 
a plane. 

But the sum of these partial departures, b a of Fig. 1, is 
evidently less than C E, the distance between the meridians 
left and arrived at on the parallel C E, which is nearest the 
equator ; and greater than B A, the distance of these meri- 
dians on the parallel Bl A, which is farthest from the equa- 



TRAVERSE SAILING. 11 

tor. But it is nearly equal to F G, the distance of these 
meridians on a middle parallel between C and A ; and 
exactly equal to the distance on a parallel a little nearer 
the pole, and whose precise position will be subsequently 
determined. (See Problem 10, Mercator's Sailing.) 
We take then i a = \ (i + i'), or, more exactly, 

Z = -%(Z + Z f ) + AZ, 

as the latitude for the departure, p. 

6. Middle Zatitude Sailing regards the departure, p, as 
the distance between the meridian left and that arrived at 
on the middle parallel of latitude ; or takes Z = \ (Z + Z'). 

T R AT ERSE SAILING. 

7. If the ship sail on several courses, instead of a single 
course, she describes an irregular track, which is called a 
Traverse. 

Problem 3. To reduce several courses and distances to a 
single course and distance, and find the corresponding dif- 
ferences of latitude and departure. 

Solution. If in Fig. 1 we regard C as different for each 
partial triangle, and represent the several courses by 6\, (7 2 , 
<7 3 , &c, we evidently have 



l x =± d x cos Ci, 




p x = d x sin dj 


l % = d 2 cos C 2 , 




p 2 = d 2 sin (7 8 , 


l z = d z cos <7 3 , 




p z — c7 3 sin <7 3 , 


&c. 




&c. 


[*] = ?!+ k+ k> &c, 


l>] 


= l>i + J P* + J ft> &c, 



and 

or, as in the more simple case of a single course, 

The ivhole difference of latitude is equal to the sum of the 
partial differences of latitude ; 

The whole departure is equal to the sum of the partial de- 
partures. 



12 



NAVIGATION. 



This applies to all cases, if we use the word sum in its 
general or algebraic sense. 

If we represent byZ n the sum of the northern diffs. of latitude, 
" " L s " " southern " M 

" P xo " " western departures, 

" P e " " eastern 

we have as the arithmetical formulas, 

[ I ] = L n ~ L s of the same name as the greater, 
[p] = P w ~I> e « 

which accord with the usual rules. (Bowd., p. 59 and p. 
264.) 

The Traverse Form (p. 60 and pp. 266 to 286) facilitates 
the confutation. 

The course, (7, and distance, [c?], corresponding to [/] and 
[j:>], may be found nearly by Plane Sailing.* 

8. The departure may be regarded as measured on the 
middle parallel, either between the extreme parallels of the 
traverse, or between that of the latitude left and that ar- 
rived at. In a very irregular traverse it is difficult to deter- 
mine the precise parallel ; but, except near the pole, and for 



* C and [d~\ are not accurately found, because [_p], the sum of the partial 
departures of the traverse, is not the same as p y the departure of the loxo- 
dromic curve connecting the extremities of the traverse. Thus suppose a 
ship to sail from C to A by the traverse 
C B, B A, her departure will be by tra- 
verse sailing d e + m n ; whereas, if the 
ship sail directly from C to A, the depart- 
ure will be o p, which is greater or less 
than d e + m n, according as it is nearer to, 
or farther from the equator. Thus we 
should obtain in the two cases a different 
course and distance between the same 
two points. In ordinary practice, how- 
ever, such difference is immaterial. 




PARALLEL SAILING. 



13 



a distance exceeding an ordinary day's run, the middle lati- 
tude suffices. (Bowel., p. 59, note.) 

It is easy, however, to separate a traverse into two or 
more portions, and compute for each separately. 



PARALLEL SAILING. 



9. The relations of the quantities (7, d, Z, and p are ex- 
pressed in equations (3). When the difference of longitude 
also enters, then some further considerations are necessary. 

Problem 4. To find the relations between 

Z, the latitude of a parallel, 

p, the departure of two meridians on that parallel, and 

D, the corresponding difference of longitude. 

Solution. In Fig. 3, let 

PAA',PC C be two meridians. 
AC = jp, their departure on the 

parallel A C, whose latitude is 

A O A' = -O A B = Z, and whose 

radius is B A = r. 
A' C = Z>, the measure of A P C, 

the difference of longitude of the 

same meridians, on the equator 

A' C' a whose radius is O A' = A = JR. 

A C, A! C are similar arcs of two circles, and are there- 
fore to each other as the radii of those circles ; that is, 

A C : A' C = B A : O A', or p:D = r: JR. 

In the right triangle O B A, 

B A = O A X cos O A B, or r = R cos Z; (4) 

that is, the radius of a parallel of latitude is equal to the 

radius of the equator multiplied by the cosine of the latitude. 

Substituting (4) in the preceding proportion, we obtain 




14 



NAVIGATION. 



p : D = cos L : 1, 



or 



p — D cos Z, D — p sec X, (5) 

which express the relations required. (Bowd., p. 63.) 

These relations may be graphically represented by a right 
plane triangle (Fig. 4), of which 



I) is the hypothenuse, 

L, one of the angles, 

p, the side adjacent that angle. 



The Traverse Table, or Table of Right Triangles, may 
therefore be used for the computation (Bowd., p. 65, u by 
inspection"). 




MIDDLE LATITUDE SAILING. 

10. Problem 5. Given the course and distance and the 
latitude left, to find the difference of longitude. 
Solution. By plane sailing, 

I z=z d cos 6 T , p = d sin C; (3) 

by Arts. 2 and 6, 

JJ = Z + l, Z Q '= J (Z'+Z) = Z+} l ; (6), 

and by equation (5), 

D=psecZ , (7) 

or JD — d sin (7 sec Z . (8) 

Equations (3), (6), and (7) or (8) afford the solution re- 
quired. 

The assumption of Z = -J (X' + i), or the middle latitude, 
suffices for the ordinary distance of a day's run; but for 
larger distances, and where precision is required, we must 
take (Art. 5) 



MIDDLE LATITUDE SAILING. 



15 



in which A L is a small correction to be added numerically 
to the middle latitude. A formula for computing it is 
given in Prob. 10, under "Mercator's Sailing." Its value in 
the most common cases is given in Bowd., p. 76, and in 
Stanley's Tables, p. 338. 

11. Strictly, the middle latitude should be used only when 
both latitudes, L and Z', are of the same name, as is evi- 
dent from Fig. 1. 

If these latitudes are of different names, and the distance 
is small, \ (L + L'), numerically, may be used; or we may 
even take p = Z, since the meridians near the equator are 
sensibly parallel. 

If the distance is great, the two portions of the track on 
different sides of the equator may be treated separately. 
Thus, in Fig. 5, the track 

C A is separated by the equator into two parts, C E and E A. 
For C E, we have 

p x = — Z tan (7, 
C'E^Z^ =p l seG$Z i 

= — Z tan C sec i Z 
nearly. 
For E A, we have 
A'A = ^=.2/, 

p 2 == U tan (7, 
EA' =D 2 =p 2 sec£Z', 
=Z' tan C sec ^ U 
nearly. 
Whence we obtain C A' or D — A + -A- 

Instead of the middle latitudes \ Z and -J- Z', we may use 
more rigidly (| Ii+A Z) and (|Z' + A L'). 

When several courses and distances are sailed, as is ordi- 
narily the case in a day's run, p and I are found as in trav- 




16 



NAVIGATION. 




Fig. 6. 



erse sailing, and then D by regarding p as on some parallel 
midway between the extremes of the traverse. (Art. 8.) 
(Bowd., p. 264.) 

12. The relations of the quantities involved in middle 
latitude sailing, namely, 

(7, d, p, I, X , and D, 
are represented graphically by combin- 
ing the two triangles of Plane Sailing 
and Parallel Sailing, as in Fig. 6, in 
which 

C = A C B, 

<Z=CA, 

j9 = BA, 

Z = C B, 

i = BAE, 

Z> = AE. 

By these two right triangles, all the common cases classed 
under Middle Latitude Sailing (Bowd., p. 68) maybe solved, 
if we add the formulas, 

13. Other problems may be stated, which never occur in 
practice; as, for example, — 

Problem 6. Given the course and distance, and the differ- 
ence of longitude, to find both latitudes. 

Solution. We have, c, d, and D being given, 

p = d sin C, 
I = d cos (7, 

cos Z = |, 

L={L Q —AL)— £/, 
Z'=(Z_JZ) + H 

A L being taken from the table (Bowd., p. 76) correspond- 
ing to X . 



MIDDLE LATITUDE SAILING. 17 



Examples in Middle Latitude Sailing. 

L and \ represent the latitude and longitude of the place sailed from. 
L and ?,', the latitude and longitude of the place arrived at. 

X /L It 71 COURSE. DIST. 

1. 39 30S. 74 20E. 41 28 S 70 30 E. S.W.byW. 210 

2. 46 24 N. 47 15 W. 49 15 N. 42 21 W. X. E. £ E. 270 

3. 51 10 S. 168 37 E. 48 31 S. 158 42 E 

4. 22 18 S. 57 28E E. by S. 317 

5. 23 15 S. 13 35 W E. 255 

6. 20 5N. 154 17 W. 18 28 S E. S E. $ E. ... 

7. 56 N. 29 34 W S. 47° E. 168 

8. 45 16 S. 3 46E. 43 10 S. 5 22 W 

9. 57 ION. 178 51 W N. 6fpts. E. 290 

10. Required the bearing and distance of Cape Race from 
Cape Hatteras. 

[tan C = j cos (i + J L) <&—l sec C] 



Cape Hatteras, 
Cape Race, 
I 
I* 


35 15 1ST. 

46 39 " 
= 11 24 " = 
=40 57 D = 


75 31 W. Tab.LIV. 

53 5 " 

684 , 
22 26 E. = 1346 


logD 


= 4- 17 
=41 14 
3.1290 




lcos(Z + d Z) 9.8762 

ar co log I 7.1649 

<7=N.55°57'E.?.tan<7 0.1701 


log I 2.8351 
1. secC 0.2518 


tf=1222' 




log d 3.0869 



Note. — The logs of I and D may be obtained from the Table of " loga- 
rithms of small arcs in space or time " in the American Ephemeris and in 
Chauvenet's Lunar Method by regarding ' and * as ° and '. 

11. A ship sails from Cape Frio south-easterly until her 



18 NAVIGATION. 

departure is 3173 miles, and then, by observation, is in lati- 
tude 34° 30' S. ; required the course, distance, and longi- 
t tude. 

[tan C—t d=lsecC D — p sec (i + A Z)] 

O / O / 

Cape Frio, 23 1 S. 41 59 W. 

U — 34 30 « p — 3173 E. logp 3.5015 

I— 11 29 " logZ 2.8382 

Z = 28 45 " <7:=S. 77 45 E. L tan (7 0.6633 

JI= 17 " 1. sec (7 0.6733 

d — 3247 logce 3.5115 

Z + AZ= 29 2" 1. sec 0.0583 

D — 61 54 E. log D 3.5698 
A' = 19 55 E. 

12. A ship in latitude 39° 8' N., longitude 33° 45' W., 
sails N". 51° 5' E. 1014 miles ; required her position. 

Z' = 49° 45' N. r = 15° 16' W. 

13. A ship in latitude 56° 46' S., longitude 170° 0' E., sails 
E. N. E. until she is in latitude 50° 10' S.; what is the dis- 
tance sailed, and what is her longitude ? 

d — 1035 miles, X ! == 163° 9' W. 

14. A ship in 42° 42' N., 12° 49' W., sails 645 miles N. Wy., 
and is then in 49° 30' 1ST. ; required the course and longi- 
tude in. 

C = N. 50° 46' W. A' = 24° 51' W. 

15. A ship sails from Port Jackson in New-Holland N. 
40° W. until the departure made is 300 miles ; what is her 

position ? 

Z= 33° 50' S. Z'= 27° 52' S. 

X = 151° 18' E. A' =c 145° 28' E. 

.16. A ship in latitude 18° 50' N., and longitude 153° 45' 
W., sails S. E. | E., 3656 miles; what is her position? 
This example conies under Art. 11. 



MIDDLE LATITUDE SAILING. 19 

\l = d cos G -A = L tan (7 sec (| L + A L ) 

D 2 =: U tan C sec \% E + A Z')] 

C = S. 41 E. 1. cos C 9.8024: 1. tan C 0.0858 

d— 3656' logtf 3.5630 log L 3.0531 

J = 38° 40' S. log I 3.3654 1. sec (U + J i) 0.0080 

Z=18 o 50'N. D 1 = 23°22'E. log A 3,1469 

L = 19° 50' S. 1. tan C 0.0858 

iL+JL= 9° 25'+ 1° 31'= 10°56'N. log L 3.0756 

■iX'+J-^'= 9° 55'+ 1° 37'= 11° 32' S. 1. sec Q £'+ A L) 0.0089 
i) i = 23°22'E. D a = 24°40'E. log D 2 3.1703 

D == 48° 2' E. 
A = 153° 45' W. 
7J = 105° 43' W. 

If J i and ^ i r are neglected, the resulting value of U 
will be 105° 57' TV". If the computations are made with the 
middle latitude, 0° 30 S., // will be 106° 39' W., or in error 
nearly 1°. 

17. Find the latitudes of two places, whose longitudes 
are 12° 49' W. and 24° 51' W., their distance 645 miles, and 
the course from the first to the second N". 50° 46' W. 
(Problem 6.) 



c = 


N. 50 46 W. 




1 


cos 


9.8010 




1. sin 9.8891 


d = 


645 






log 


9.8096 




log 9.8096 


l=z 


6 48 N. 






log 


9.6106 




log^ 9.6987 


z> = 


12 2 W. 

46 12 N. or 


S. 










log 9.8585 


4 = 






1. cos 9.8402 


a r=x 


— 6 
















46 6 tf. 




or 




46° 6' 


S. 




**= 


3 24 X. 








3 24 


N. 




L- 


42 42 N. 




or 




49 30 


S. 




U = 


49 30 X. 




or 




42 42 


s. 





Examples in Traverse Sailing. 

A ship from the position given at the head of each of the 
following traverse forms sails the courses and distances 



20 



NAVIGATION. 



stated in the first two columns ; required her latitude and 
longitude. 

1. August 8, noon— Lat. by Obs., 35° 35' N. 
Long, by Chro. 18° 38' W. 



Courses. 


DlST. 


N. 


S. 


E. 


W. 


N. N. E. i E. 


/ 

50 


44.1 


/ 


23.6 


' 


S.|W, 


46.2 




45.7 




6.7 


S. by E. i E. 


16.5 




15.8 


4.8 




N. E. 


38 


26.9 




26.9 




S. S. W. i w. 


41.8 




37.8 




17.9 




192.5 


71.0 


99.3 


55.3 


24.6 


S. 4J E. 


41.5 




28.3 


30.7 
38 = 


= D. 



August 9, noon — Lat by Acct., 35° 7' IS. 
Long. " 18° 0' W. 

2. September 25, noon— Lat. by Obs., 49° 53' S. 

Long, by Acct., 158° 27' E. 



Courses. 


DlST. 


N. 


s. 


E. 


w. 


Pts. 


/ 


/ 


' 


/ 


/ 


S. 4* E. 


45.3 




28.7 


35.0 




S. 5£ E. 


19.5 




10.0 


16.7 




S. 7 W. 


38 




7.4 




37.3 


S. 6i W. 


25.7 




8.7 




24.2 


S. 3 W. 


51.2 




42.6 




28.4 


N. 7i E. 


13 


1.9 




12.9 




N. 5f E. 


10 


4.3 




9.0 
73.6 






202.7 


6.2 


97.4 


89.9 


S. 1 W. 


93 




91.2 


D = 


16.3 
= 26 



September 26, 8 a.m. — Lat. by Acct., 51° 24' S. 
Long. " 158° 1' W. 

In this example the courses are expressed m points, which 
is the preferable method. 

When the reductions are the same for all the compass 
courses, we may find the difference of latitude and depar- 



MERCATOR S SAILING. 



21 



ture for these compass courses, and the course and distance 
made good. The traverse is thus referred to the mag- 
netic meridian instead of the true. The course made good 
may then be corrected for variation, etc. ; and with this cor- 
rected course and the distance made good the proper diifer- 
ence of latitude and departure may be found. 



3. September 16, 6 p.m.- 



-Lat by Obs., 
Long, by Chro. 



50° 16\S. 



Comp. Course. 


DlST. 


N. 


s. 


E. 


w. 


S. W. i s. 


25 


' 


19.3 


' 


15.9 


s. s. w. 


30 




27.7 




11.5 


S. by W. 


18 




17.7 




3.5 


s. 


43 




43 






S. by E. i E. 


255 




24.7 


6.2 




S. E. i S. 


33 




26.5 


19.7 






174.5 






25.9 


30.9 


(map.) S. 2° W. 


159 




158.9 




5.0 


Var'n, &c, 18° W. 












(true) S. 16° E. 


159 




152.8 


43.8 




or S. by E. £ E. 








70 = 


= D 



September 17, noon — Lat. by Acct., 52° 49' S. 
Lon^. " 



MERCATOR' S SAILING. 

14. Middle Latitude Sailing suffices for the common pur- 
poses of navigation ; but a more rigorous solution of pro- 
blems relating to the loxodromic curve is needed. These 
solutions come under " Mercator's Sailing." 

Pkoblem 5. A ship sails from the equator on a given 
course, C, till she arrives in a given latitude, X, to find the 
difference of longitude, D. 

Solution. Let the sphere (Fig. 7) be projected upon the 
plane of the equator stereographically. The primitive circle 
A B C . . . ,M is the equator. 



22 



NAVIGATION. 




Fig. 7. 



P, its centre, is the pole (the eye or projecting point being 
at the other pole).* 

The radii, PA, PB, PC, &c, are meridians making the 
same angle with each other in the projection as on the sur- 
face of the sphere.* 

The distance P m, of any point 
m from the centre of the projec- 
tion, =tan|(90 o — X), the tangent 
of \ the polar distance of the 
point on the surface which m re- 
presents, the radius of the sphere 
being 1.* 

This curve in projection makes 
the same angle with each merid- 
ian, as the loxodromic curve with 
each meridian on the surface.* 

A M is the whole difference of longitude D. 

If we suppose this divided into an indefinite number of 
equal parts, A B, B C, C D, &c, each indefinitely small, and 
the meridians P A, PB, PC, &c, drawn, the intercepted 
small arcs of the curve A b c .... m may be regarded as 
straight lines, making the angles P A 5, P b c, P c d, &c, 
each equal to the course C ; and consequently the triangles 
PAJ,PJc,Pcrf, &c, similar. 

We have then 

PA:PJ = PS:Pc = Pc:P(?, &c, 
or the geometrical progression, 

PA:PJ :Pc:....Pra. 

If then 

_D = the whole difference of longitude, 
d = one of the equal parts of 2>, 

~j will be the number of parts, and 

-j + 1 the number of meridians PA,P5....Pm, 
* Principles of stereographic projection. 



mercator's sailing. 23 

or the number of terms of the geometrical progression : and, 

employing the usual notation, 

the first term a = PA = l, 

the last term l — Ym — tan £(90° —Z), 

the ratio r s= ^— . 

PA 

To find this ratio, we have in the indefinitely small right 

triangle AB5, 

tan BAJ = cotPAJ=:^-r, 

or „ PA — Vh 

cot 6 = 5 , 

a 

whence P A— P 6 == d cot (7/ 

Pft = PA-dfcot C, 
and, since PA=I, 

P5 
r = p-: = l — df cot (7. 

Then by the formula for a geometrical progression, 

(Algebra, p. 240,) we have 

tan \ (90°— Z) = (l — c? cot C) 7 . (8) 

Taking the logarithm of each member, we have 

logtan|(90°— Z)=^log(l— tfcot C). (9) 

But we have in the theory of logarithms 

71/ 71? 71/ 

(JVaperian) log (1 + ^)=^ — — + — — — + &c 

and 

|- ^,2 ^,8 ^ 4 "1 

( Common) log (1 +7i) = m rc — — + «- •— j" + & c — K 10 ) 

in which the modulus m =.434294482. 

Hence, putting n = — c? cot (7, 
log (1 -6? cot £7)= m [-d cot (7-i d 2 cot 2 C-\ d 2 cot 3 <7-&c. . . .], 
and substituting in (9) and reducing, 

log tan i (90°— Z)= -m x D [cot tf+£ cZ cot 2 G 

+ id 3 cot 3 6'+&c. ...]. C 11 ) 



24 



NAVIGATION. 



This equation is the more accurate the smaller d is taken, 
so that if Ave pass to the limit and take e?=0, it becomes 
perfectly exact. The broken line A b e. . . .m then becomes 
a continuous curve, and our equation (11) becomes 

log tan \ (90°— L) = — m X D cot C ; 
whence 



2> = 



logtani(90°-Z) 



tan (7, 



(12) 



But in this equation D is expressed in the same unit as 
tan (7, that is, in terms of radius. (Trig., Art. 11.) 

To reduce it to minutes we must multiply it by the radius 
in minutes, or r'=3437'.74677. 

Substituting the value of m, we shall have (in minutes), 

3437'. 74677, . - /aa0 rw n 
.484294482 l ^ Un * < 90 ~ Z > tan (7 ' 



Z> = - 



To avoid the negative sign, w^e observe that 

1 1 



tan £ (90°— X): 



cot i(90°-Z) tan £(90° + Z)' 



or that 



—log tan i (90°— Z)=log tan £ (90° + i). 

Hence we have, by reducing, 

Z> = 7915 , .70447 log tan (45° + £Z) tan C. (13) 

Note. — Problem 5 may be more readily solved, and equation (13) obtained 
by aid of the Calculus. 

In Fig. 1, suppose c a to be an element, 
or infinitesimal part, of the loxodromic 
curve C A : 

cb will be the corresponding element of 

the meridian, and 

b a x sec X, the element of the equator ; 

L being the latitude of the indefinitely 

small triangle cab. 

Fig. 1. 




mercator's sailing. 25 

By articles 5 and 10, using the notations of the Calculus, we have 
d X = cos Odd d p = tan C d X 
d D = sec Ldp = tan (7 sec XdX, 
in which C i3 constant. 

By integrating the last equation between the limits X = and X = Z, we 
shall have 

r L 
X = tan C\ sec X d X, 

the whole difference of longitude required in Problem 5. 

To effect the integration, put 

sin X = x, then by differentiating 

d x 

d X = — , and multiplving by sec L 

cos L 

t j t- d x d x 

sec X d L = 



sec Z d X = 

1 — x 



cos 2 X 1— sin 3 X' 
d x 



Resolving into partial fractions, we obtain 

sec X d X = 4- — v and 

Ll + x 1 — xj 

J o L secXdX = i[log(l +x)-log(l-x] 
= log|/L^ 

' 1 — X 

. . /l+sin 

= iogy - — - 

" 1 — sic 



X 

-sinX 
-sinX 

= log tan (45° + 1 X) Trig. (154). 

Whence we have 

D = log tan (45° + i X) tan C. 
But in this the logarithm is Naperian, and D is expressed in terms of the 
radius of the sphere. To reduce to common logarithms, we divide by 
,m =.434294482, and to minutes by multiplying by r'= 3437'.T4677, and 
obtain 

D = 1915'. 70447 log tan (45° + } X) tan C y 
as in (13). 

15. To facilitate the practical application of the formula 
just obtained, put 

ilf=7915 , .70447 log tan (45° + £ L) ; (14) 



26 



NAVIGATION. 



and let M be computed for each minute of L from up- 
ward, and its values given in a table. We shall thus form 
the Table of Meridional Parts or of Augmented Latitudes, 
such as Bowditch's Table III. This formula accords with 
that given in the Preface. (Bowd., Pref. p. iv.) 

In practice, then, we have only to take the value of M 
corresponding to X, and D is then found by the formula, 

D=M tan C. (15) 

M has the same name, or sign, as L. 

Example. 

To find tne meridional parts, or augmented latitudes, for 
each minute, from 30° to 33° ; 

log 791 5'.70447= 3.898490. 

L. 45° + JZ. log tan. I. log tan. log M. M. 



30 
30 20 

30 40 

31 
31 20 

31 40 

32 
32 20 
32 40 



60 
60 10 
60 20 
60 30 
60 40 

60 50 

61 
61 10 
61 20 



0.2385606 
.2414830 
.2444154 
.2473580 
.2503108 
.2532741 
.2562480 
.2592328 
.2622286 



9.377599 

.382887 
.388129 
.393326 
.398480 
.403591 
.408660 
.413690 
.418680 



3.276089 
.281377 
.286619 
.291816 
.296969 
.302081 
.307150 
.312180 
.317170 



1888.37 

23.14 
1911.51 7 

23.21 

1934.72 8 
23.29 

1958.01 9 

23.38 
1981.39 7 

23.45 
2004.84 9 

23.54 
2028.38 9 

23.63 
2052,01 9 

23.72 

2075.73 8 
23.80 

2099.53 



33 61 30 .2652356 .423632 .322122 

The second differences afford a check of the work. 
By interpolating into the middle, M can be found for each 
10' ; and then, by simple interpolation, for each 1'. In the 
first step, one eighth of the second difference is to be sub- 
tracted. The following is an example : 



hercator's sailing. 27 

M L M 



30 






1888.37 


30 11 


1901.09 


1 


1889.53 


12 


1902.24 


2 


1890.68 


13 


1903.40 


3 


1891.84 


14 


1904.56 


4 


1893.09 


15 


1905.72 


6 


1894.15 


16 


1906.87 


6 


1895.31 


17 


1908.03 


1 


1896.46 


18 


1909.19 


8 


1897.62 


19 


1910.35 


9 


1898.77 


20 


1911.51 





1899.93 




&c. 



16. Problem 6. A ship sails from a latitude, L, to an- 
other latitude, U, upon a given course, C / find the differ- 
ence of longitude, D. 
Solution. Let 

M be the augmented latitude corresponding to L, 
M' " " " " JO. 

The difference of longitude from the point, A, where the 
track crosses the equator to the 1st position, whose latitude 
is L, will be 

D =31 tan C; 

and to the second position, whose latitude is U, 

D=M' tan C ; 
and we shall have 

D=D-I>={M'-M) tan C; (16) 

or, when M r <M, 

&=D i -D M =(M-M') tan C; 

since the sign of D is determined by the course. 

If L and L' are of different names, so also are JSfand M ', 
and we have numerically 

D=(M+jr) tan C. 

17. The difference, M'—M, is called the meridional, or 



28 



NAVIGATION. 




Fig. 8. 



augmented, difference of latitude. Representing this by m y 
we have 

D = m tan C. 

The relation between these quantities _ 
is represented by a plane right triangle 
( Fi g- 8 )> in which 

C is one of the angles, 

m = CE, the side adjacent, 

D = E F, the side opposite. 

The triangle of " Plane Sailing" has 
the same angle (7, with 

;— C B, the adjacent side, 
and ]) = BA, the opposite side. 

Fig. 8 represents these two triangles combined. By them, 
all the common cases under Mercator's Sailing can be solved, 
either by computation or by the Traverse Table. (Bowd., 
p. 79.) 

The relations between the several parts involved are 
I — d cos (7, L'=L + l, 

p^dsmC, m =M'—M, 

D — m^nC, X f ~X + D; 

and since p = 1 tan (7, 

I : m =p : D. 

18. Problem 7. Given the latitudes and longitudes of 
two places, find the course, distance, and departure. (Bowd., 
p. 79, Case I.) 

Solution. L and U being given, we take from Table III. 
JfandJf'. 

We have I = Z'-Z, m = M'-M, D ^ X'- X- 

by Mercator's sailing, tan C = — ; 

and by Plane sailing, d = I sec (7, p = I tan (7/ 



(18) 



mercator's sailing. 29 

/, m, and C are north or south according as U is north or 
south of X. 

D, p. and (7 are east or wes£, according as X' is eas£ or 
westf of A. 

If the two places are on opposite sides of the equator, we 
have numerically 

Mercator's sailing is rarely used except in this case, and 
when the differences of latitude and longitude are consider- 
able. 

There are two limits of its accuracy :— 

1. Table III. contains the augmented latitude only to the 
nearest minute or mile.* 

• 2. It is computed on the supposition that the earth is a 
sphere. Some works on Navigation, as Mendoza Rios and 
Riddle, contain a table of augmented latitudes, in which 
the true form is taken into consideration.! 

Examples. 
1. Required the course and distance from Cape Frio to 
34° 30' S., 18° 30' E. 



Cape Frio, 


23 1 S. 


41 59 W. 


M - 1420 S. 




£'== 


34 30 S. 


W = 18 30 E. 


M* = 2208 S. 


log D 3.5598 


I = 


11 29 S. 


D = 60 29 E. 


M = 788 S. 


logm 2.8965 


c =s. 


11 45 E. 


1 sec C 
log/ 


0.6733 

2.8382 


1. tan C 0.6633 


d = 


3247' 


logd 


3.5115 





* The most convenient unit for nautical distances is the geographical, 
nautical, or sea mile, which is 1' of the earth's equator, or 6086.43 feet. 
Regarding the earth as a sphere, this is also 1' of any great circle, 
f The formula for the terrestrial spheroid is 

M— 7915'.70447 log tan (45° +£ L) 

— 22'.98308 sin Z + 0'.01276 sin 3 L+ &c. 
Delambre has shown that a table of meridional parts constructed for the 
sphere may be used for the spheroid by using as the argument the geocen- 
tric latitude instead of the true latitude. 



30 NAVIGATION. 

2. Required the course and distance from Cape Frio to 
Lizard Point, England. 



Cape Frio, 23 1 S. 




41 59 W. 


M 


= 1420 S. 




Lizard Pt., 49 58 N. 




5 12 W. 


M' 


— 3471 N. 


log D 3.3438 


1= 72 59 N. 


Dr 


= 36 47 E. 


M 


= 4891 N. 


log m 3.6894 


<7=:N.24 17 E. 




1. sec C 
log* 




0.0402 
3.6414 


1. tan C. 9.6544 


d = 4804' 




logd 




3.6816 





3. A ship in latitude 18° 50' 1ST., longitude 153° 45' W., 
sails S. 4^ points E., 3656 miles ; what is her position ? 

, , d = 3656 log d 3.5630 

L = 18 50 N. A = 153 45 W. M = 1151 JST. 1 cos C 9.8024 

I = 38 40 S. log I 3.3654 

L' — 19 50 S. M' — 1215 S. 1. tan C 0.0858 

M — 2366 S. log m 3.3740 

log D 3.4598 



D - 48 3 E. 
X = 105 42 W. 



19. Other problems might be stated than those com- 
monly given ; as, for example, — 

Problem 8. Given the latitude left, the course and both 
longitudes, to find the latitude arrived in. 

Solution. We have D = A'— X, 

by Mercator's sailing m —D cot C (N. or S. as is C), 

by Table III. M corresponding to L, 

M' = M+m, 
and again by Table III, U corresponding to M'. 

Problem 9. Given the difference of longitude and differ- 
ence of latitude of tico places, and the course between them, 
find both latitudes. 

Solution. We have 

m = M— M — D cot C. 
But 3F= 7915'.70447 log tan £ (90° + Z') 

M = 7915'.70447 log tan £ (90° + Z), 



mercatok's sailing. 31 

consequently, 

log tan \ (90° +27) - log tan * (90° +L) = 79 f 5 ^ 7 - (19) 

Put logcot^^^, * (20) 

then equation (19) gives 

tani(90° + Z') , . 

t^-WTT) = cot *• 

By PL Trig. (109) 

tan i (x + y) sin x -f sin y 

tan \ (x — y) sin a; — sin y' 

In this, if we take 

x + y = 90° + L' 

x — y — 90° + X, ' 



we have 
or putting 



cc = 90 o +i (Z'+i), 



i = | {11 + L) the middle latitude, 
»== 90° + i , 
and y = £ (Z'-Z)-= J Z, 

and 

tan j (90° + Z') _ cos L Q + sin j- Z __ 

tan" J (90° + L) ~~ cos Z — sin £ 2 ~~ COt " 

whence 

T cot ^ + 1 . , f 
cos Jj = — — sin 4- /, 

cot 9 — 1 z ' 

which, by PL Trig. (151), reduces to 

cos X = tan (45° + 0) sin £ I. (21) 

We have also 

The solution is effected by equations (20), (21,) (22). 



32 



NAVIGATION. 



Example. 



The difference of longitude of two places is 5 10 E. 



3 28 N". 
N. 32 59 E; 

ar. co. log 6.10151 

log 2.49136 

log cot 0.18776 

log 8.78063 

log tan 1.15900 
log sin 8.48069 
W cos 9.63969 



the difference of latitude, 
the course 
find the latitudes. 
(Constant) 7915'.70447 

C= 32° 59' 
log cot = 0.06034 
0=41° 2' 
45°+ = 86° 2' 
%l = 1° 44' 

Z =s 64° 8' N". or 64° 8' S. 
Z = 62° 24' 1ST. or 65° 52' S. 
L' = 65° 52' N. or 62° 24' S. 
This problem cannot be solved with precision when Z is 
near 0. 

20. Problem 10. To find the correction of the middle 
latitude in Mid, Lat. Sailing. (Tab. Bowd., p. 76 ; Stan- 
ley, p. 338.) 

Solution. In Mid. Lat. Sailing we have 

cosZ = -g. (23) 

in which precision requires that Ave take 
Z = i(Z' + Z) + AZ; 
A Z being a correction of the middle latitude, which it is 
now proposed to find. 
In plane sailing p = I tan C, 

in Mercator's sailing D = ?n tan (?, 

which substituted in (23) give 

l_ 
■m 



COS Z : 



(24) 



MERCATOR'S SAILING. 33 

whence 1—2 sin 2 i i ==— , 

and sin £ i = |/ m ~ Z . (25) 

Now for different values of Z and X' we may find 
(in minutes) I = U — X, 
the middle latitude, i m == ^ (X' + Z), 
m=7915'.70447 [log tan (45° + | Z') -log tan (45° + £ Z)],* 
and then i by (24), or if small by (25), from which sub- 
tracting Z m we have A Z, which is required. 

In computing m, logarithms to 7 places should be used 
when the difference of latitude is less than 12°. 

The correction of the Middle latitude computed for differ- 
ent middle latitudes and differences of latitude may be given 
in a table, as on page 76 (Bowd.) It becomes too large to 
be conveniently tabulated, when the latitudes are of differ- 
ent names, or the middle latitude is very small. 

Example. 

Find the correction of the Middle Latitude, when the lati- 
tudes are i = 12°, Z ! = 18°. 

o / 

(45° + £ Z') = 54 I. tan 0.1387390 

(45° -f | Z) — 51 I. tan 0.0916308 const, log 3.89849 

i {Z' + Z) = 15 0.0471082 log 8.67310 

I = 6 == 360' ar. co. log 7.44370 

Z — 15 7 I. sec 0.01529 

AZ = 7 

21. The loxodromic curve on the surface of the earth and 
its stereographic projection (Fig. 7) present a peculiarity 

* log 7915.70447 = 3.8981896. Another formula, requiring only 5 place 
logarithms, is 

m = 6875'.493 [q + 1 ? 3 + \ q* + \ q 1 + - .] 
in which q = sin ^ 1 sec % (L' +L). 



34 NAVIGATION. 

worthy of notice. Excepting a meridian and parallel of 
latitude, a line which makes the same angle with all the me- 
ridians which it crosses would continually approach the 
pole, until, after an indefinite number of revolutions, the 
distance of the spiral from the pole would become less than 
any assignable quantity. It is usual to say that such a curve , 
meets the pole after an infinite number of revolutions. Still, 
however, it is limited in length. 

For we have for the length of any portion, 
by plane sailing, d == (Z'—Z) sec C. 
If Z =: o and Z' = 90° = J, 

the whole spiral from the equator to the pole will be, 
with radius == 1, 

d =— sec C. 

If Z = - 90° = -J, and Z' = 90° =J, 

we have, as the entire length from pole to pole, 

d = re sec C, 
If also C =: 0, or the loxodromic curve is a meridian, 
d = 7r, a semicircumference, as it should be. 

So also the length of the projected spiral A b c . . . (Fig. 1) 
from A to m can readily be shown to be (calling this 
length 6) ; 

6 = Mm sec C = [1 — tan (45° — £ Z)] sec (7, 

or, (H. Trig. (.51),) * = ^f^; 

and its length from the equator to the pole — taking Z == 90° ; 

d = sec C. 

a meecatoe's chart. 
22. On a Mercator's chart, the equator and parallels of 
latitude are represented by parallel straight lines ; and the 



A MERCATOR S CHART. 



30 



meridians also by parallel straight lines at right angles with 
the equator. Two parallels of latitude, usually those which 
bound the chart, are divided into equal parts, commencing 
at some meridian and using some convenient scale to repre- 
sent degrees, and subdivided to 10', 2', 1', or some other 
convenient part of a degree, according to the scale em- 
ployed. 

Two meridians, usually the extremes, are also divided 
into degrees and subdivided like the parallels of latitude, 
but by a scale increasing constantly with the latitude : so 
that any degree of latitude on such meridian, instead of be- 
ing equal to a degree of the equator, is the augmented de- 
gree, or augmented difference of 1° of latitude, derived from 
a table of " meridional parts." (Bowd., Table III.) The 
meridian is graduated most conveniently by laying off from 
the equator the augmented latitudes ; or from some parallel, 
the augmented difference of latitude for each degree and 
part of a degree, — using the same scale of equal parts as 
for the equator. 

Thus, on such a chart, 

the length of 1° in lat. is 60' of the equator, 



IC 


cc 


cc 


10° 


cc 


61' 


cc 


cc 


cc 


cc 


cc 


20° 


cc 


64' 


cc 


cc 


cc 


cc 


cc 


30° 


cc 


69' 


cc 


cc 


cc 


cc 


cc 


40° 


cc 


78' 


cc 


cc 


cc 


cc 


cc 


50° 


cc 


93' 


cc 


cc 


cc 


cc 


cc 


60° 


cc 


120' 


cc 


cc 


cc 


cc 


cc 


70° 


cc 


176' 


cc 


cc 



<fcc, 



and the augmented difference of latitude 

from 0° to 10° is 603' = 10° 3' of the equator, 
cc 10 o u 2o° " 622' = 10° 22' " " 

" 20° " 30° " 663' = 11° 3' " " &c. 

23. As on other maps and charts, parallels of latitude and 



36 NAVIGATION*. 

meridians are drawn at convenient intervals ; places, shore 
lines of continents and islands, harbors and rivers, &c, are 
plotted, each point in its proper position ; and such configu- 
rations of the land represented as the purpose of the map 
requires. 

To plot on a chart a point, whose latitude and longitude 
are given, — by means of the scales at the sides, draw a pa- 
rallel of latitude in the latitude, and by means of the scales 
at the top or bottom, a meridian in the longitude of the 
point ; or so much of each as suffices to find their intersec- 
tion. (Bowd., p. 88.) 

In nautical charts the soundings in shoal water are put 
down, and even the character of the bottom ; and on those 
of a large scale, also, the contour lines of the bottom, or 
lines of equal depth. The variation of the compass at con- 
venient intervals, and lines of equal variation, are valuable 
additions. 

24. The meridians on this chart being parallel, arcs of 
parallels of latitude are represented as equal to the cor- 
responding arcs of the equator : thus each is expanded in 
the proportion of the secant of its latitude to 1 ; as is evi- 
dent from the formula 

JD —p sec L, 

It can be shown that very small portions of the meridians 
are expanded in the same proportion. This is apparent from 
the table of the length of 1° in Art. 22; as for example, a 
degree whose middle latitude is 60° is 120', or, 

60' of the equator X sec 60°. 
But the two half degrees are unequally expanded ; for 
from 59£° to 60° is represented by 59', 
" 60° to 60£° " " " 61', nearly. 

A small circle on the surface of the earth of 1° diameter, 



A MERCATOR'S CHART. 



87 



at the equator is then represented by a circle, whose diame- 
ter is 1° ; 
in lat. 30° nearly by a circle, whose diameter is l°Xsec 30°, 

" 60° " " " " l°Xsec60°, 

L « " V " l°Xseci; 

but not exactly by a circle, since the meridians are aug- 
mented more rapidly as the latitude is greater. 

Such a chart, then, while representing a narrow belt at the 
equator in proper proportions, presents a view of the earth's 
surface expanded at each point, both in latitude and longi- 
tude, proportionally to the secant of its latitude. 

25. If Ave take any two points, C F, on this chart, and 
join them by a straight line, and form a right triangle by 
a meridian through one, and a parallel of latitude through 
the other, we shall have the triangle of Mercator's sailing 
(Fig. 8) : for, the intercepted portion of 
the meridian, C E, is the augmented dif- 
ference of latitude ; and of the parallel 
of latitude, E F, is the difference of longi- 
tude. Hence the angle E C F is the course. 
(Art. 17.) Moreover, the loxodromic 
curve is represented by the straight line 
C F ; for if we take any intermediate 
point of this curve, and let d be its posi- 
tion on the chart, d must be in the line 
C F, otherwise when we construct the 
triangle of Mercator's sailing, we shall 
have an angle at C different from E C F, the course ; which 
for every point of the loxodromic curve is the same. 

Thus a Mercator's chart presents two decided advantages 
for nautical purposes, viz. : 

1. The ship's track is represented by a right line. 

2. The angle, which this line makes with each meridian, is 
the course. 




Fig. 8. 



38 NAVIGATION. 

To find the course from one point to another on the chart, 
all that is necessary is to draw a line, or lay down the edge 
of a ruler, through the two points, and measure its angle 
with any meridian. A convenient mode is to refer such line 
by means of parallel rulers to the centre of one of the com- 
pass diagrams, which usually will be found on the chart, and 
reading the course from the diagram. Another mode of 
transferring the line to the compass diagram is described on 
page 88. (Bowd.) 

As such diagrams, except on some charts of limited ex- 
tent, are constructed with reference to the true meridian, the 
course obtained is the true course, and not the compass 
course. 

26. The distance, C F, however, is an augmented distance, 
which we may measure nearly by the augmented scale on 
the meridians of the chart (the middle latitude of the scale 
used being the same as that of the line C E). (Bowd., p. 88.) 
Or we may construct the proper distance, C A, by construct- 
ing the triangle, C B A, of plane sailing, in which C B is the 
proper difference of latitude, the scale for which is on the 
equator. 

The distance here spoken of, though represented on this 
chart by a straight line, is not the shortest distance between 
the two points, — for on the surface of a sphere, the shortest 
distance between two points is the arc of a great circle, 
w r hich joins them. To find this belongs to great-circle 
sailing. 

GREAT-CIRCLE SAILING. 

27. The rhumb-line, or spiral curve, which cuts all the 
meridians at the same angle, has been used mostly by navi- 
gators in passing from point to point on account of the sim- 
plicity of the calculations required in practice. But, as has 
been stated, it is a longer line than the great circle between 
the same points, and therefore the intelligent navigators of 



GREAT-CIRCLE SAILING. 



39 



the present day are substituting the latter wherever practi- 
cable. 

On the Mercator chart, however, the arc of a great circle 
joining two points, not on the equator or on the same meri- 
dian, will not be projected into a straight line, but into a 
curve longer than the Mercator distance, and still greater 
than the distance on a rhumb-line. Hence it is an objection 
to the Mercator chart, that the shortest route from point to 
point appears on it as a circuitous one ; and this is, doubt- 
less, one main reason why merely practical men have made 
so little use of the great circle. Many of those unacquainted 
with the mathematical principles of the subject are unable 
to comprehend how the apparently circuitous path on their 
chart should actually be the line of shortest distance. 

28. Problem 11. To project on a chart the arc of a great 
circle joining two given points on the globe. 

Solution. It will be necessary to project a number of 
points of the arc, and trace through these points the curve 
by hand. To project a point on the chart, we must know 
its latitude and longitude. 

The two given points, A and B 
(Fig. 9), and the pole, P, are the 
three angular points of a spheri- 
cal triangle, formed by the arcs 
joining these points with each 
other and with the pole. If from 
P we draw P C perpendicular to 
A B, the point C is nearer the 
pole than any other point of A B ; 
that is, it is the point of maximum 
latitude. This point of greatest latitude is called the vertex 
of the great circle. 

1st. To find the latitude and longitude of this vertex. 

This may be done by a direct application of the rules of 
spherical trigonometry, first finding the angles A and B by 




Fig. 9. 



40 NAVIGATION. 

Case I. of Sph. Trig., and then solving one of the right tri- 
angles A JP C 01 B P C . But in practice the following 
method is preferable. 

Let X 1 = (90°— P A), the latitude of A, or the less latitude, 
Lcf=z (90°— P B), the latitude of B, or the greater latitude, 

A= A P B, the difference of longitude of B from A. 
Z„=(90°— P C ), the latitude of the vertex. 
Aizz: A P C , the longitude of the vertex from P A. 
A 2 =B P C , the longitude of the vertex from P B. 
The right triangle A P C , gives 

a __ tan P C ___ cot L v tan L x . . 

' l tan PA cot L x tan L v ' ^ ' 

and the triangle B P C , 

tan P C cot L v tan Z 2 



cos A 2 = 



tan P B cot Z 2 tan L v ' 

Whence by division 

cos \ tan L x 
cos A 2 tan Z 2 ' 

and by composition and division, 

cos \ — cos \ tan Z 2 — tan L x 

cos A 2 -f- cos \ tan X 2 -f tan L x 

By PI. Trig. (110) and (126), this equation becomes 
tan * (A, + A,) tan \ (A, - \) = g°^g. 

But A =z (X x -f- A 2 ), and if we put A A = % (A x — A 2 ), we have 

tan A A = ^#-=^ cot i A 5 (27) 

sin (Z 2 + Z x ) z ' v ' 

A x = £ A + A A, (28) 

A 2 = £A — AX. (29) 

By (27) we find A A, and then A x by (28), which, applied 

to the longitude of A, or the point wiiose latitude is the 

smallest, and in the direction toward B, gives the longitude 

of the vertex. 



GREAT-CIRCLE SAILING. 



41 



For finding the latitude of the vertex, equations (26) give 
tan Z v = tan Z l sec A n ) , n 

tan Z v = tan i 2 sec A 2 , f 
either of which may be used. 

In using (27) attention must be paid to the signs of Z x and 
Z 2 . If the greater latitude, Z 2 , is regarded as positive, £„ 
when of a different name is negative ; and in this case 
(i 2 — X L ) will be numerically the sum, and (Z 2 + Z 1 ) the dif- 
ference of the two latitudes. In this case we shall find 
X x > 90°. 

When A X > % A, X 2 is negative, and the vertex, C , is be- 
yond B, as in Fig. 10, instead of being between A and B, as 
in Fig. 9. 

In (30) we have Z v positive, or of the same name with the 
greater latitude, since numerically X 2 < 90°. 

The vertex, which is here used, is that which is nearest 
the point B, or the place whose latitude is numerically the 
greater. For this in (27) A X < 90°. 

There are, however, two vertices, which are diametrically 
opposite, as C and C' of the great circle C' E C in Fig. 10. 
For the vertex C, we have in (27) A X> 180°, or in the 
third quadrant, and in (30), Z v of a different name from Z 2 . 

2d. To find any number of points, C, C", C"\ &c, C 19 C 2 , 
C 3 , &c, we may assume at pleasure the differences of longi- 
tude from the vertex C P C, C P C", C P G", &c. It is 
best to assume them at equal intervals of 5° or 10°. 
Let X ! = C P C, U = (90°- P C), the lat, of C, 

a" = C PC", Z^^-PC"), " C", 
X'"=: C P C", Z m = (90°- P C ,,; ), " C"\ 

&C. &G. 

then the right triangles C P C r , C P C'\ C P C", &c, give 
tan U = tan L v cos X\ 

tan L" = tan Z v cos X", \ (31) 

tan Z'"= tan Z v cos X'", &c. J 



42 



NAVIGATION". 



Or we may assume values of Z', Z", L"\ &c., and find the 
corresponding values of A/, A", A'", &c., by the formulas 

cos X' z=z tan J7 cot L v 

cos A" =i tan L" cot i y h (32) 

cos A"'= tan X'" cot Z v &c. J 

from which we shall have two values of A for each value 
of Z. 

Having thus found as many points as may be deemed 
sufficient, we may plot 
them upon the chart, and 
through them trace the 
required curve. 

29. Another method con- 
sists in finding the longi- 
tude of the point E (Fig. 
10), where the great circle 
intersects the equator, and 
then by the right triangles 
E C c\ E C" c", E &' c"\ 
&c, the latitudes and lon- 
gitudes of C, 0", C", &c. 
Let X L = E a, the longitude of A from E ; 
A 2 = E 5, the longitude of B from E ; 
A = X 2 — X x = a&, the difference of longitude of B 

from A ; 
Li and Z 2 , respectively the latitudes of A and B ; 
i v — Q C , the latitude of the vertex, and also the 
measure of QE C , the inclination of the 
great circle to the equator. 
From the right triangles E A a, E B S, we have 

tan A a tan B b 




Fig. 10. 



or 



tan E = 
tan L v = 



sin E a 
tan L x 



* sin E V 
tan Z.> 



sin I, sin 



7 * 



GREAT-CIRCLE SAILING. 



43 



whence 



sin ?i t 
sin /t 2 



tan L x 



tan L 2 ? 
and by composition and division, 

sin ^ + sin \ tan L 2 + tan Z x 

sin X 2 — sin a x tan L 2 — tan Z x * 

By PL Trig., (109) and (126), this becomes 

tan £ (% 2 + /Ij) sin (Z 2 4- L x ) 

tan \ {\ — \) sin (L. 2 — L x )* 

But A = A 2 — A x , and if we put A'X = -| (A 2 -}- A x ) we have 

sin (Z 2 4- A) 



tan J 'A 



sin (L 2 — L x ) 

A x = J'A ■ 

A 2 = J'A + iA 



tan £ A 



£A 



(33) 



a 



(34) 



From these we find A 19 or A 2 , which applied to the longitude 
of A, or of B, gives the longitude of E, the intersection with 
the„ equator. This point, it may be observed, is nearer A 
than B ; and is outside of A, when the two places are on the 
same side of the equator ; and between A and B when they 
are on different sides. 

For finding the inclination of the circle to the equator, we 

have above 

tan L v = tan i x cosee A x 

tan L v = tan X 2 cosee A 2 
in which L v will have the same name, or sign, as Z 2 . 

To find any numbers of points, C, C", C", &c, we may as- 
sume at pleasure, as in Art. 28, the differences of longitude, 
A', A", A'", &c, from E ; and from the right triangles E CV, 
E C" c" E C" c"\ &c, find the corresponding latitudes 
L\ U\ U'\ &c, by the formulas 

tan U = tan L v sin A' 
tan Z"= tan L v sin A" j> (35) 

tan Z"= tan L v sin A'" &c. J 
The great circle intersects the equator at two opposite 



44: NAVIGATION. 

points. The intersection, E, given by these formulas is that 
which is nearest A, the place whose latitude is the smallest. 

This method is preferable to that of Art. 28, only when 
the two places are on different sides of the equator, and the 
intersection with the equator is between them. In this case, 
X l and A 2 , as well as L^ and Z 2 , will have opposite signs. 

30. Problem 12. To find the great-circle distance of two 
given points. 

Solution. Let A and B (Fig. 9) be the two given points. 

In the triangle P A B are given, as in Problem 11, 
PA = 90 o -Z n P B = 90°-Z 2 , A P B = A> 
to find AB = c?, the distance required. 

1st Method. By (27), (28), (29), and (30), we may find 
Z„, the latitude of the vertex, and X x and A 2 , the longitudes 
of the two given places from the vertex. 

Then from the right triangles, A C P, B C P, (Fig. 9), 
putting A C = d i and BC = c? 2 , we have 
tan c?! = tan X x cos L v 
tan d 2 = tan A 2 cos L v \ (36) 

and d = d Y + d 2 . 

d reduced to minutes Avill be the distance in geographical 
miles. 

When A 2 is negative, which happens when the vertex is 
beyond B (Fig. 10), d 2 is also negative, and d is numerically 
the difference of d l and d 2 . 

2d Method. By (33) and (34) we may find X l and A 2 the 
longitudes of the two given places from E (Fig. 10), the in- 
tersection, and L v the angle which the circle makes with the 
equator. 

Then from the right triangles, E a A, E b B, we have, 
putting d f = E A, d"= E B, 

tan dl = tan X v sec L v 1 
tan d"~ tan A a sec L v \ (37) 

and d=d"-d'. J 



GREAT-CIRCLE SAILING. 45 

When £ x and L 2 are of different names, so also are a x and A 2 , 
and d is numerically the sum of d' and d" . 
Zd Method. By Sph. Trig. (4) we have 

cos d c= sin j^ sin Z 2 + cos X x cos L 2 cos A, 

and, putting 

Jc cos = sin X 2 ) _ 

* sin 4> = C os A cos X or tan ^ = cot A cos A, 1 

COS 



, sin Z 2 sin (Z 2 + 0) 
COS a = 2 » 



from which <f) and then c? may be found. <f> is in the same 
quadrant with X. 

30. Problem 13. To find the course on a great-circle 
route. 

Solution. If A (Fig. 9), the point whose latitude is nu- 
merically the smaller, is the point of departure, it is required 
to find the angle P A B : if B is the point of departure, then 
the angle P B A. 

1st Method. The position^ of the vertex having been found 
by (27), (28), (29), (30), we have from the right triangles, 
A C P, B C P, 

cos A = sin L v sin X x ) 

cos B == sin L v sin A 2 j ^ ' 

in which A < 90° ; and B < 90°, when the vertex is between 
A and A (Fig. 9), but > 90°, when the vertex is beyond B 
(Fig. 10). 

2c? 3fethod. Having found the intersection and angle with 
the equator by (33) and (34), we have from the right tri- 
angles E a A, E b B (Fig. 10), 



cos A = sin L v cos X x 
cos B == — sin L v cos JL 



Sd Method. By Napier's Analogies we have 



46 



NAVIGATION. 



tan £ (B + A) 
tan i (B-A) 



cos \ {L^—L x ) 
sin i (Lz+LJ 
sin \ (A— L x ) 



cos \ (Z 2 + L x ) 
A=i(B + A)-l(B 

B = * (B + A) + | (B 



cot J A 

cot % X 
A) 



A) 



(41) 



When A and B are on opposite sides of the equator, 
^ (i 2 — i x ) is numerically half the sum, and % (J0 2 + X x ) is 
half the difference of the two latitudes. 

31. When the courses are found by this last method, the 
distance may be found by 



or, 



tan £ d •■ 



tan £ d : 



sin j(B + A) 
: sin i (B— A) 

cos^(B + A) 
: cos i (B-A) 



tan $ (i 3 — ij), 



cot i (i 2 +ii), 



(42) 



The 1st is preferable, when % (Z 2 -f .Z^) is near 0, and conse- 
quently J (B-f A) is near 90° ; the second when % (X 2 — L x ) 
and consequently \ (B — A) are near 0. (Sph. Trig. 74.) 

32. Example. To find the great circle from San Francisco 
to Jedo. (Formulas 27, 28, 29, 30.) 



San Francisco, 
Jedo, 



Lat. Z 2 = 37 48 N. 

L x = 35 40N. 

Z 2 +Z 1= 73 28 

Z a — L x = 2 8 

A = 97 38 

i A = 48 49 

JA= 1 57 

\=\X—A\= 50 46 

^=£ A— .J A = 46 52 



Long 122 22 W. 
140 E. 
1. cosee 0.0183 
1. sin 8.5708 

1. cot 9.9420 

1. tan 8.5311 
1. sec \ 0.1990 L sec ^0.1651 
1. tan L x 9.8559 1. tan Z 2 9.8897 



Vertex, Z„=48° 37' N. Long.169 14 W. I. tan Z„ 0.0549 0.0548 



GREAT-CIRCLE SAILING. 



47 



Long, from " 

Vertex. L cos A. 1. tan L. 




± 5 
±10 
±15 
±20 
±25 
±30 
±35 
±40 
±45 
±50 



0.0000 
9.9983 
.9933 
.9849 
.9729 
.9573 
.9375 
.9134 
.8842 
.8494 
.8081 



0.0549 
.0532 
.0482 
.0398 
.0278 
.0122 

9.9924 
.9683 
.9391 
.9043 
.8630 



Lat. 

48 37 N. 
48 30 
48 10 
47 37 
46 50 
45 48 
"44 30 
42 55 
41 
38 44 
36 7 



Longitudes. 



169 14 W. 
164 14 
159 14 
154 14 
149 14 
144 14 
139 14 
134 14 
129 14 
124 14 
119 14 W. 



169 14 W. 

174 14 
179 14 W. 

175 46 E. 

170 46 
165 46 
160 46 
155 46 
150 46 
145 46 
140 46 E. 



(Vertex). 



(36). 



(39) 



1. sin A x 
1. sin A 2 
1. sin L v 
1. cos d 



9.8891 
9.8632 
9.8753 
9.7444 



1. cos C» 9.7385 



1. tan A x 0.0880 
1. tan A 2 0.0283 
L cosi v 9.8202 
1. tan d x 9.9082 
1. tan d 2 9.8485 
d, = 39° 0' Course, C l N. 54° 27' E. from JTedo. 
d 2 = 35 13 C 9 N. 5Q 48 W. from San Francisco. 

Distance, d = 74° 13' = 4,453 miles. 
Distance by Mercator's sailing, 4,689 miles. 

33. To follow a great circle rigorously requires a con- 
tinual change of the course. As this is difficult, and indeed in 
many cases is practically impossible, on account of currents, 
adverse winds, &c, it is usual to sail from point to point by 
compass, thus making rhumb-lines between these points. 

When the ship has deviated from the great circle which 
it was intended to pursue, it is necessary to make out a new 
one from the point reached to the place of destination. It 
is a waste of time to attempt to get back to an old line. 

34. As the course, in order to follow a great circle, is 
practically the most important element to be determined, 
mechanical means of doing it have been devised. Towson's 
Chart and Table is used much by English navigators. 



48 



NAVIGATION. 



Chauvenet's Great-Circle Protractor renders it as easy as 
taking the rhumb-line course from a Mercator chart. 

Charts have been constructed by a gnomonic projection, 
on which great circles are represented by straight lines ; 
but by these computation is necessary to find the course. 

35. A great circle between two points near the equator 
or near the same meridian differs little from a loxodrornic 
curve. But when the differences both of latitude and of 
longitude are large, the divergence is very sensible. It is 
then that the great circle, as the line of shortest distance, is 
preferred. 

But it is to be noted, that in either hemisphere the great- 
circle route lies nearer the pole, and passes into a higher 
latitude, than the loxodrornic curve. Should it reach too 
high a latitude, it is usually recommended to follow it to the 
highest latitude to which it is prudent to go, then follow 
that parallel until it intersects the great circle again. 

36. A knowledge of great-circle sailing will often enable 
the navigator to shape his course to better advantage. Let 

A B (Fig. 11) be the loxodrornic curve on a Mercator's chart, 
AC B, the projected arc of a great circle. 

The length on the globe of the great circle A C B is less 
than that of the rhumb-line A B, or of any other line, as 
A D B, between the two. But A C B is also less than lines 
that may be drawn from A to B on 
the other side of it, that is, nearer 
the pole ; and there will be some 
line, as A D' B, nearer the pole 
than the great circle, and equal in 
length to the rhumb-line. Between 
this and the rhumb-line may be 
drawn curves from A to B, all less 
than the rhumb-line. If the wind 
should prevent the ship from sail- 




Fig, n. 



SHAPING THE COURSE. 49 

ing on the great circle, a course as near it as practicable 
should be selected. If she cannot sail between A B and 
A C, there is the choice of sailing nearer the equator than 
A B, or nearer the pole than A C. The ship may be near- 
fing the place B better by the second than by the first, al- 
Ithough on the chart it would appear to be very far off from 
the direct course. 

This may be strikingly illustrated by the extreme case of 
a ship from a point in a high latitude to another on the same 
parallel 180° distant in longitude. The great-circle route is 
across the pole, while the rhumb-line is along the small 
circle, the parallel of latitude, east or west ; the two courses 
differing 90°. Any arc of a small circle drawn between the 
two points, and lying between the pole and the parallel of 
latitude, will be less than the arc of the parallel. Hence 
the ship may sail on one of these small circles nearly west, 
and make a less distance than on the Mercator rhumb, or 
parallel due east. This is, indeed, an impossible case in 
practice, but it gives an idea of the advantage to be gained 
in any case by a knowledge of the great-circle route. 

It is possible in high latitudes that a ship may have such 
a wind as to sail close-hauled on one tack on the rhumb- 
line, and yet be approaching her port better by sailing on 
the other tack, or twelve points from the rhumb-line course. 

37. The routes between a number of prominent ports 
recommended by Captain Maury are mainly great-circle 
routes, modified in some cases by his conclusions respecting 
the prevailing winds. 

SHAPING THE COURSE. 

. 38. The intelligent navigator, in selecting his course to a 
destined port, will not only have regard to the directness 
of the route, but will take into consideration obstructions 
and dangers which may be in his way ; prevailing winds 



50 NAVIGATION. 

and currents ; and, in case of a threatening storm, the course 
to be taken to avoid its greatest violence, or being driven on 
a lee shore. 

Good charts and books of sailing directions afford all 
requisite information respecting obstructions and dangers 
in the most frequented seas. Exploring expeditions from 
England, France, and the United States have of late years 
added greatly to this branch of nautical knowledge. 

The labors of Maury, and his recent colaborers in Eng- 
land and France, have greatly increased our knowledge of 
prevailing winds in large portions of the ocean. The care- 
ful observations of intelligent navigators are much needed 
still further to develop it. 

A few of the stronger currents, such as the Gulf Stream 
off the coast of the United States, are well known. But 
more extended observations are wanted. Currents are often 
indicated by the difference in the temperature of their wa- 
ters from that of those surrrounding them ; so that the 
thermometer, as well as barometer, has become an import- 
ant instrument to the navigator. 

The works of Redfield, and especially of Reid and Pid- 
dington, afford much information respecting storms and tor- 
nadoes. That class of storms called cyclones is particularly 
deserving of attention. . 

These branches of physical geography are well worthy 
of study by those engaged in navigating the ocean. 



CHAPTER II. 

REFRACTION. —DIP OF THE HORIZON.— PARALLAX.— 
SEMIDIAMETERS. 



REFRACTION. 

39. It is a fundamental law of optics, that a ray of light 
passing from one medium into another of different density 
is refracted, or bent, from a rectilinear course. If it passes 
from a lighter to a denser medium, it is bent toward the 
perpendicular to the surface, which separates the two me- 
dia ; if it passes from a denser to a lighter medium, it is 
bent from that perpendicular. Let 

M and N (Fig. 12) represent two 
media each of uniform density, 
but the density, or refracting 
power, of N being the greater; 

a b c, the path of the ray of light 
through them ; 

P 5, the normal line, or perpendicu- 
lar, to the separating surface at b. 

If a J is the incident ray, b e is the refracted ray ; P b a 
is the angle of incidence / P b a 1 is the angle of refraction. 

If c b is the incident ray, b a is the refracted ray, and 
P b a! and P b a are respectively the angles of incidence 
and refraction. 

Moreover, these angles are in the same plane, which, as it 
passes through P 5, is perpendicular to the surface at which 
the refraction takes place ; and we have for the refraction 




Fig. 12. 



52 



NAVIGATION. 



a r b a ~ P b a — P b a\ 
or the difference of direction of the incident and refracted 
rays. 

A more complete statement of the law for the same two 
media is, that 

- — TJrnr-7 — m > a constant for these media ; 
sin V o a 

or, the sines of the angles of incidence and refraction are in 
a constant ratio. 

This law is also true when the surface is curved as well as 
when it is a plane. 

40. If the medium N instead of being of uniform density 
is composed of parallel strata, each uniform but varying from 

each other, the refracted ray b c will 
be a broken line ; and if, as in Fig. 
13, the thickness of these strata is 
indefinitely small, and the density 
gradually increases in proceeding 
from the surface 5, b c will become a 
| n curved line. But we shall still have 
for any point c of this curve, c a! be- 
Fig. 13. j ng a tan g ent to it, 

sin P h a 

a constant for the particular strata in which c is situated. 

This law, which is true for strata in parallel planes, ex- 
tends also to parallel spherical strata, except that the nor- 
mals P i, P' c are no longer parallel, but will meet at the 
centre of the sphere. But the refraction takes place in the 
common plane of these two normals. 

41. The earth's atmosphere presents such a series of pa- 
rallel spherical strata, denser at the surface of the earth, and 
decreasing in density, until at the height of fifty miles the 
refracting power is inappreciable. 




REFRACTION. 



53 



In Fig. 14, the concentric, circles M N" represent sections 
of these parallel strata, formed by the vertical plane passing 
through the star S and 

the zenith of an observer Z E S 

at A. The normals C A Z 
at A, and C B E at B, 
are in this vertical plane. 
S B, a ray of light from 
the star S, passes through 
the atmosphere in the 
curve B A, and is re- 
ceived by the observer 
at A. 

Let A S' be a tangent 
to this curve at A ; then Fig. 14. 

the apparent direction of 

the star is that of the line A S' ; and the astronomical re- 
fraction is the difference of directions of the two lines B S 
and A S'. This difference of directions is the difference of 
the angles EBS,EDS', which the lines SB,S'A make 
with any right line C B E, which intersects them. If, then, 
r represent the refraction, we have 

r = E B S-E D S'. 

Also, E B S is the angle of incidence, and Z A S', the appa- 
rent zenith distance, is the angle of refraction ; and we 

have 

sin E B S 




sin ZAS' 



== m, 



a constant ratio for a given condition of the atmosphere and 
a given position of A ; but varying w T ith the density of the 
atmosphere, and for different elevations of A above the sur- 
face. For a mean state of the atmosphere and at the sur- 
face of the earth, experiments give m = 1.000294. 

The principles of Arts. 39 and 40, applied to this case, 



54 NAVIGATION. 

show that astronomical refraction takes place in vertical 
planes, so as to increase the altitude of each star without 
affecting its azimuth. The refraction must therefore be 
subtracted from an observed altitude to reduce it to a true 
altitude; or 

h = h'-r, 
in which h is the true altitude, 

h\ the apparent altitude, 
r, the refraction. 
These laws are here assumed. The facts and reasoning 
on which they depend belong to works on optics. (Bowd., 
p. 153 ; HerschePs Astronomy, p. 37 ; Lardner's Optics.) 
42. Problem 14. To find the refraction of a star. 
Solution. In Fig. 14, let 

2 = Z A S', the apparent zenith distance of the star, 

r = EBS-EDS',the refraction, 

w=ZCE; 

then EDS' = ADC = ZAS'-ZCE^2-w, 
E B S % E © S' f r w a — u + n 

, sin E B S sin (z — u 4- r) 

and . „ . Q , = = m, 

sm ZAS sin z 

or sin \z— (u— r)~\ == m sin z\ (43) 

which is of the same form as (309) of Plane Trig., 

sin (z + a) = m sin z, 
the solution of which gives 

tan (z + ia) he tan * a. 

Putting a =z — (ii—r), its value in (43), we have 

tan [z-i {u-r)] = j— J tan J (u—r) ; 



whence 



tan | (u-r) = y-j-^ tan [z— £ (w-r)]. (44) 



REFRACTION. 



55 



In this u and r are both unknown, but we may note that 
each is a very small angle, being when the zenith distance 
is 0, and increasing with the zenith distance. As it is ne- 
cessary to make some supposition respecting them, let us 
assume that they vary proportionally, and that 

u 

7 = ?' 

a constant, reserving it for observations to test the rigor of 
this assumption. 

Equation (44) then becomes 

tan \ (q-1) r = ~=^ tan [>-£ (y-l>r], 

or, since i {q— 1) r is quite small, 

* (3.-1) r sin 1" =\=^ tan |>-| (q-1) r] ; 



whence 






1 +m 



1 — m 



tan (z— \ (q— 1) r). 



(q — 1) sin V 1+m 

Since observation is to determine q, we may as well con- 
sider that it determines the whole of the constant factor 
into which q enters. 

2 1 — ra 



Put then 



n = ■ 



(q — 1) sin 1" 1+m' 

and the formula reduces to 

r — n tan (z— p r), 

which is known as Bradley's formula. 

Suppose at two given zenith distances z' and z n the refrac- 
tions r' and r ,r are found by observations in a mean state of 
the atmosphere, then we have the two equations, 

r' = n tan (z! —p r'), 
r /r = ntan ty'—pr"); 



56 • NAVIGATION. 

and the two unknown quantities n and p may be found by 
proper transformations, or by successive approximations. 

By comparing pairs of observations in this way at various 
zenith distances, the values of n and p come out very nearly 
the same, except at very low altitudes ; so that the hypothe- 
sis that <7, and therefore n and p are constant, is found to 
be nearly, though not rigorously exact. 

The values that have been found are, with the barometer 
at 29.6 inches, and the thermometer at 50°, 

n ~ 57 /; .036, p = 3 ; 

and the formula by which Tab. XII. (Bowd.) has been com- 
puted is 

r =z 57".036 tan (z — 3 r). (46) 

In computing by this formula, we must find an approximate 
value of r, by assuming first 

r = 57".036 tan z, 
and substitute the value thus obtained in the second member 
of the proper equation. 

Example. 

Find the refraction for the altitude 30°. 

log 57".036 — 1.75615 . . . . . . . . 1.75615 

2=60° l.tanO.23856 z— 3 r = 59°55'4" 1. tan 0.23712 

r= 98".8 log 1.99471 r = 98 /r .46 log 1.99327 

3r = 4' 56".4 r = l'38 /r .46 

43. Laplace, from a more profound investigation of this 
problem, obtained a more complicated formula, which agrees 
better with observations. 

Bessel has modified and improved Laplace's formula. His 
tables of refractions are now considered the most reliable. 
They are found in a convenient form for nautical problems 
in Chauvenet's Method of Equal Altitudes, Table III. The 
mean refractions in this Table are for the height of the 



REFRACTION. 57 

barometer, 30 inches, and the temperature 50° of Fah- 
renheit.* 

44. The mean values of n and p in Art. 42 correspond to 

the height of the barometer, b = 29.6 inches, 
the thermometer, t = 50° Fahrenheit. 

Xow, the refraction in different conditions of the atmos- 
phere is nearly proportional to the density of the air ; and 
this density, the temperature being the constant, is propor- 
tional to its elasticity ; that is, to the height of the baro- 
meter. Hence, if 

b is the noted height of the barometer (in inches), 
r, the mean refraction of Tab. XII., 
A r, the correction for the barometer, 
r -\- A r ~b 



then 



29.6' 



By this formula the correction for the barometer in Tab. 
XXXVI. is computed. 

Again, the elastic force being constant, the density in- 
creases by T £o part for each depression of 1° Fahrenheit. 
Hence, if 

A 1 r = the correction for the thermometer, 

t = the temperature in degrees of Fahrenheit, 

A ' r = ^m^( r + A ' r ) ( 48 ) 

which reduces to A 1 r = ^ a0 . ^ r 

oOU -f- t 

* Chauvenet's Astronomy, Vol. I. pp. 127-172, contains a thorough in- 
vestigation of the problem of refraction, especially of Bessel'? formulas. 



58 



NAVIGATION. 



by which the correction for the thermometer in Table 
XXXVI. is computed. 

Bessel's formulas are more rigid, but more complex. 

45. Problem 15. To find the radius of curvature of the 
path of a ray in the earth? s atmosphere. 

Solution. By the radius of curvature for any point of a 
curve, is meant the radius of the circular arc, which most 
nearly coincides with the curve at that point. 

If we consider the curvature of the path of a ray to be 
uniform from B to A (Fig. 14), it is the same as considering 
the curve B A itself to be a circular arc, and the problem is 
reduced to finding the radius of this arc. 

Let C be the centre of the arc A B, 
B! ~ C A, the radius of curvature, 
H = C A, the radius of the earth. 



Z E S 




Fig. 14. 



Since S B and S' A are 
tangents to the curve at 
B and A respectively, 
they are perpendicular 
respectively to the radii 
C B, C A; hence, 

AC'B = r, 

the difference of direc- 
tions of S B and S' A. 

As A B is a very small 
arc, we may put 



AB=:i2' sinr, 



and, since they are very nearly equal, we may also put 
AD=AB=:jft'sinr. 



REFRACTION'. 



59 



In the triangle ADC, 

AD 

AC" 

R' sin r 



sin A C D 

sin ADC 



or 



whence 



R 



sin (z — u) ' 
R sin u 



sin (z — u)' sin r 
or nearly enough, since u and r are small, 

B! = — - 5. 

sin s* r" 
But in the preceding problem 

u = qr, jP = i^— 1)== 3 > 

whence 2—^f u = 7 r ; 

7R 



so that 



J?'= 



sin 3' 



(49) 



(50) 



(51) 



which is the required formula, nearly. 

46. When z = 0, or the star is in the zenith, 

JR = cc ; 

that is, the path is a straight line. 

When z = 90°, or the object is in the horizon, 

H r = 1 E ; 

that is, near the earth's surface a ray of light nearly horizon- 
tal moves in a curve, which is nearly the arc of a circle whose 
radius is seven times the radius of the earth. 

This, however, is in a mean condition of the atmosphere. 
The curve is greatly varied in extraordinary states of the 
atmosphere, or by passing near the earth's surface of differ- 
ent temperatures ; in very rare cases even to the extent of 
becoming convex to the surface a short distance. 



60 



NAVIGATION 



DIP OF THE HORIZON. 




Fig. 15. 



47. Problem 16. To find the dip of the horizon. 

Solution. Let A (Fig. 15) be the position of the observer 
at the height BA = A, above the 
level of the sea ; A H, perpendic- 
ular to the vertical line, C A, re- 
presents the true horizon. 

The most distant point of the 
horizon visible from A is that at 
which the visual ray, H" A, is 
tangent to the earth's surface. 

The apparent direction of H" 
is A H', the tangent to the curve 
A H" at A. /H == H A H' is 
the dip of the horizon to be 
found. 

Let C be the centre of the earth, 

C, the centre of the arc H" A. 

H", C, C, are in the same straight line, since the arcs H" B, 

LT A are tangent to each other at H", 

C A, C A, are perpendicular respectively to AH, A H' ; 

hence 

C A C'= II A H'= A H, the dip. 

Let JR = C B, the radius of the earth ; 

then R + h = C A, 

7 B = C A = C B.% the radius of curvature of H"A, 

We have, then, in the triangle C A C, by PL Trig. (268), 
siniJH^j/^ 



• **)(**) . 



7 B.(R + h) » 

and, since h is comparatively very small and may therefore 
be omitted alongside of i?, 



DIP OF THE HORIZON. 



61 



or, putting sin \ A H—\ A H sin 1", 

AH= * i/Jl=*^/J-Vh. (52) 

sin 1" V 7 E sin 1" r 7E 

Taking B = 20923596 feet (Herschel, p. 126), we find the 
constant factor 



2 J 8 = 59".040, 
sinl" ' 7i2 



(53) 



and 



J,ff=59 ,/ .040 4 / A, 
log J j? = 1.77115 + | log h, 

h being expressed in feet, which is nearly the formula for 
Tab. XIII. (Bowd.) 

Since . „ y — ^ is constant, depending only upon the 

radius of the earth, J H is proportional to Vh y or the dip is 
proportional to the square root of the height of the observer 
above the level of the sea. 

48. Were the path of the ray, H A, a straight line, we 
should have 

A f H=KAW=WCA, 



and in the triangle H /; C A 

cos A' Hz 



E 
JK+ff 



whence, 
and 



2 sinVj#2?=^=|, nearly, 



AJECz 



h 
2J2' 



sin I' 

or with h in feet, A' H= 63".77l Vh. 

Comparing this with A H= 5 9". 040 Vhl we find 

A H= AH- 4".731 Vhzzz A'lT-.OU A' H y 



(54) 



62 NAVIGATION. 

or that the dip is decreased by refraction by .074, or nearly 
tV of it. 

But from the irregularity of the refraction of horizontal 
rays (Art. 46), the dip varies considerably, so that the tabu- 
lated dip for the height of 16 feet can be relied on ordinarily 
only within 2'. When the temperatures of the air and water 
differ greatly, variations of the dip from its mean value as 
great as 4' may be experienced. In some rare cases, varia- 
tions of 8' have been found. 

The dip may be directly measured by a dip-sector. A series 
of such measurements carefully made, and under different 
circumstances, both as to the height of .the eye, temperature 
and pressure of the atmosphere, and temperature of the 
water, are greatly needed. 

Prof. Chauvenet (Astron. L, p. 176) has deduced the fol- 
lowing formula, which it is desirable to test by observations : 

t — t n 



or 



in seconds, AH- A'H-24:02l" • 



in minutes, AH— A'H—Q'.6l 



in which 



A'H 



t is the temperature of the air, 
t that of the water, 
by a Fahrenheit thermometer. 

When the sea is warmer than the air, the visible horizon 
is found to be below its mean position, or the dip is greater 
than the tabulated value ; when the sea is colder than the air, 
the dip is less than its tabulated value. (Raper's N~av., 
p. 61.) 

This uncertainty of the dip affects to the same extent all 
altitudes observed with the sea horizon. 

49. Near the shore, or in a harbor, the horizon may be ob- 
structed by the land. (Bowd., p. 155.) The shore-line may 
then be used for altitudes instead of the proper horizon. Tab. 



DIP OF THE HORIZON. 



63 



XVI. (Bowd.) contains the dip of such water-line, or of any 
object on the water, for different heights in feet and dis- 
tances in sea miles. It is computed by the formula 



q X 

D=:-d+ 0.56514 -j 
7 a 



(55) 



in which 



h is the height in feet ; 
c?, the distance of the object in sea miles ; 
Z>, the dip in minutes. 
50. Problem 17. To find the distance of an object of 
known height, which is just visible in the horizon. 

Solution. If the observer is 
at the surface of the earth at 
the point H /; (Fig. 15), a point 
A appears in the horizon, or is 
just visible, when the visual ray 
A W just touches the earth at 
H". Let 

h = B A, the height of A, 
d = H" A, the distance of A. 

As this arc is very small, we 
have 

d= H" C A sin 1' X C A = 1 R X H" C A sin l\ 
since by (51) C A = 1 B. 

From the three sides of the triangle C C A by PL Trig. 

(268), 




Fig. 15. 



sin iH'C'A = f 



/\h(R+\K) 



42 R 2 



or nearly 



and 



i H" C A sin 1 



W; 



84 £? 



H' C' A sin 1' = i/III. 

y 21 r 



64 NAVIGATION. 

This, substituted in the expression for c?, gives 

*-**VS«y (■£•**); (») 

In this, <#, A, and J? are expressed in the same denomina- 
tion. 

But if h and B are in feet, 

in statute miles, d = -^— -^ j/ f — B A J, 

in geographical miles, d = -^ |/ f- B h). 

Taking B ■== 20923596 feet as before, we find 

in stat. miles <#= 1.323 V\ orlogc?=:0.12l72-h£log A, ) , ,v 
ingeog. " c7= 1.148 Vh, or log c?= 0.05 994 + i log A. ) 

The first of these is nearly the formula given by Bow- 
ditch for computing Table X. (Bowd., Preface.) 

51. Were the visual ray, H" A, a straight line, we should 
have from the right triangle C H" A y 



H"A = |/(C A 2 -ir'C 2 ), or d f = V {2 B + h) h; 
or nearly d' = V 2 B X VL 

Introducing the same numerical values as before, we have 
in statute miles 

d' = 1.225 Vh. 

Comparing this with the expression above, we see that 
the distance is increased about y 1 ^ part by refraction. This, 
however, is subject to great uncertainty. 

52. If the observer is also elevated at the height of B' A' 
(Fig. 16), and sees the object A in his horizon, then its dis- 
tance is 

A' ir -fir a, 



PARALLAX. 



65 




or the sum of the distances of each 
from the common horizon, H". 

By entering Table X. with the 
heights of the observer and the ob- 
ject respectively, the sum of the cor- 
responding distances is the distance 
of the object from the observer. The 
distances in this table are in statute 



5280 



miles. Multiplying them by 
to geographical miles. 



=s .86751, reduces them 



P AE ALL AX . 

53. The change of the direction of an object, arising from 
a change of the point from which it is viewed, is called 
parallax ; and it is always expressed by the angle at the 
object, which is subtended by the line joining the two 
points of view. (Hersch. Ast., Art. 70.) Thus in Fig. 17, 
the object S would be seen from A in the direction A S ; 
and from C in the direction C S. The angle at S, subtended 
by A C, is the difference of these directions, or the parallax 
for the two points of view, C and A. 

54. In astronomical 
observations, the ob- 
server is on the surface 
of the earth ; the con- 
ventional point to which 
it is most convenient to 
reduce them, wherever 
they may be made, is the 
earth's centre. (Hersch. 
Ast., Art. 80.) In those 
problems of practical 
astronomy which are 
used by the navigator, 
we have only to con- 




66 



NAVIGATION. 



sider this geocentric parallax, which is the difference of the 
direction of a body seen from the surface and from the cen- 
tre of the earth. It may also be defined to be the angle 
at the body subtended by that radius of the earth, which 
passes through the place of the observer. Thus, in Fig. 17, if 

C is the centre of the earth, and 
A the place of the observer, 

the geocentric parallax of a body, S, will be the angle 

S = ZAS-ZCS, 
at the body subtended by the radius C A. 

If the earth is regarded as a sphere, C A Z will be the 
vertical line through A, and will pass through the zenith Z. 
Then will the plane of C A S be a vertical plane ; 

Z A S, the apparent zenith distance of S as observed at A ; 
Z C S, its geocentric or true zenith distance ; and 
Z A S > Z C S. 

Thus we see that this parallax takes place in a vertical 
plane, and increases the zenith distance, or decreases the 
altitude, of a heavenly body without affecting its azimuth. 

55. This suffices for all nautical problems except the com- 
plete reduction of lunar distances. 
For these and the more refined ob- 
servations at observatories, the spher- 
oidal form of the earth must be con- 
sidered. Then, as in Fig. 18, the 
radius C A does not coincide with 
the normal or vertical line C A Z, 
but meets the celestial sphere at a 
point Z', in the celestial meridian, nearer the equator than 
the zenith, Z. 

We may remark here that 

A C" E, the angle which the vertical line makes with the 
equator, is the latitude of A ; and 




Fig 18. 



PARALLAX. 67 

ACE, the angle which the radius makes with the equator, 
is its geocentric latitude. 

56. Problem 18. To find the parallax of a heavenly body 
for a given altitude. 

Solution, In Fig. 17, let 
p = S, the parallax in altitude ; 

z — Z A S, the apparent zenith distance of S, corrected for 
refraction ; 
H == C A, the radius of the earth ; 

d h= C S, the distance of the body, S, from the centre of the 
earth. 
Then from the triangle C A S, we have 

sin C S A = ^-h sin CAS, 

ii? sin z .. 

or sin p = — -= — , (58) 

If the object is in the horizon as at H, the angle A H C is 
called its horizontal parallax ; and denoting it by 7r 3 we have 
from (58), or from the right triangle C A H, 

sin 7T = ~ (59) 

which substituted in (58) gives 

sin^> = sin n sin z. (CO) 

If h = 90°— z, the apparent altitude of the object, we 
have 

sin^? = sin n cos h; (61) 

or, nearly, since p and it are small angles, 

p = n cos h. (62) 

57. The horizontal parallax, tt, is given in the Nautical 
Almanac for the sun, moon, and planets. From Fig. 17 it 
is obviously the semidiameter of the earth, as viewed from 
the body. As the equatorial semidiameter is larger than 
any other, so also will be the equatorial horizontal parallax. 



68 NAVIGATION. 

This i3 what is given in the Almanac for the moon. Strictly 
it requires reduction for the latitude of the observer, and 
such reduction is made at observatories, and in the higher 
order of astronomical observations. 

58. Tables X. A and XIV. (Bowd.) are computed by for- 
mula (62). 

Table XXIX. contains the correction of the moon's alti- 
tude for parallax and refraction corresponding to a mean 
value of the horizontal parallax, 57' 30". It should be used, 
however, only for very rough observations or a coarse ap- 
proximation. 

Tab. XIX. contains the difference of 59' 42" and the com- 
bined correction of the moon's altitude for parallax and re- 
fraction. The numbers taken from this table subtracted 
from 59' 42", give the correction of an apparent altitude for 
parallax and refraction. To this may be applied the reduc- 
tion of the refraction to the actual condition of the atmos- 
phere (Art. 42). If, instead of the equatorial hor. parallax, 
we enter the table w T ith the augmented parallax of Chau- 
venet's Lunar Method (Tab. III.), we shall obtain the re- 
duction, not to the centre of the earth, C (Fig. 18), but to 
the point, C, where the normal line through A intersects 
the axis of the earth. 

Table XIX. of Bowditch was arranged especially for one 
of the Lunar methods in that work, so that the reductions 
of the distance should all be additive. 

APPARENT SEMIDIA METERS. 

5 \ The apparent diameter of a body is the angle which 
its disk subtends at the place of the observer. 

Problem 19. To find the apparent semidiameter of a 
heavenly body. 

Solution. In Fig. 19, let M be the body; 
d = C M, its distance from the centre of the earth ; 



APPARENT SEMIDIAMETERS. 



69 



d r = AM, its distance from A; 
r = MB, its linear radius or 

semidiameter ; 
5 = MCB, its apparent semi- 
diameter, as viewed 
from C ; 
J = M A B',its apparent semi- 
diameter, as viewed 
from A (B and B' are 
too near each other to 
be distinguished in 
the diagram) ; 
iJ-CA, the earth's radius. 
1st. For finding s, the right triangle C B M, gives 

sin s = y r (63) 

Were the body M in the horizon of A, or Z A M = 90°, 
its distance from A and C would be sensibly the same, so 
that the angle s is called the horizontal semidiameter. 

In (59) we have for the horizontal parallax, 

R „ R 

sin n = ? , 

which, substituted in (63), gives 




Fig. 19. 



or d : 



sin s = r- sin 7r, 
Jtc 



or nearly, since s and n are small, 



s== n n - 



(64) 



(65) 



^ is constant for any particular body, as it is simply the 
ratio of its linear diameter to that of the earth. (Hersch. 
Ast., p. 544.) 

For the moon (Hersch. Ast., p. 214), 

■^ = 0.2729, 



n.\ ( 66 ) 



70 NAVIGATION. 

s = 0.2729 7T, 
and log s = 9.43600 + log 

By this formula the moon's horizontal semidiameter may 
be found from her horizontal parallax. 

The Nautical Almanac contains the semidiameters as well 
as the horizontal parallaxes of the sun, moon, and planets. 

2d. For finding s', the apparent semidiameter as viewed 
by an observer at A on the surface of the earth, the right 
triangle A B' M gives 





sins=^. (67 


In the triangle 


CM A, 

sinMAC CM 
sin M C A ~~ A M> 


or, putting 
and 


h = 90°— Z A M, the apparent, 

h! = 90°— Z C M, the true altitude of M, 




cos h d , 
cosA'-rf" < 68 



whence, 

,, ,cos^' 
cosh 

which, substituted in (67), and by (63), gives 

. r cos h . cos h 

Bin s = -= 77 = sin s — T7, 

a cos A cosA ' 

or approximately, s' = s — =7, (69) 

by which s' may be found when s and A are known. 

Since h < A', cos h > cos A', and consequently s r > s ; 
that is, the semidiameter increases with the altitude of the 
body. The excess 

A s = «'— s, is called the augmentation. 

The moon is the only body for which this augmentation is 
sensible. 



APPARENT SEMIDIAMETERS. 



71 



60. Problem 20. To find the augmentation of the moorf s 
horizontal semidiameter. 

/Solution. From (69) we find 

A , cos h — cos h! 

A S = S 5 — 5 r . , 

cos A' 
which, by PI. Trig. (108), becomes, 



As 



2 sin j (N + h) sin \ Qi' — h) 
cos h' 



h! — h =p, the parallax ; since it is small, we may put 

2 sin \ (h' — h) = 2 sin \p =p sin 1" == 77 cos A sin 1* ; 
and, in computing so small a quantity as A s, we may take h 
for \ (A' + A), and cos A for cos h'; and then 

A s = s 7T sin 1" sin A, 



or, since (65) 



For the moon 



S =E^ 



A s — -^ 7T 2 sin 1" sin A. 



r 
E 



0.2729; then 



A s = .000001323 tt 2 sin A. (70) 

If we take 7r = 57' 20", which is nearly its mean value, we 
have 

A s = 15".65 sin A, (71) 

which agrees nearly with the formula for Tab. XV. (Bowd.) 
The augmentation may differ 2" from this mean value. 

Tab. II. of Chauvenet's Lunar Method contains this aug- 
mentation for different values of s, as well as of A, computed 
by a more precise formula. 



CHAPTER III. 

TIME. 

61. Transit. The instant when any point of the celestial 
sphere is on a given meridian is designated as the transit of 
the point over that meridian. 

62. Hour-angle. The hour-angle of any point of the sphere 
is the angle at the pole, which the circle of declination pass- 
ing through the point makes with the meridian. It is pro- 
perly reckoned from the upper branch of the meridian, and 
positively toward the west. It is usually expressed in hours, 
minutes, and seconds of time. The intercepted arc of the 
equator is the measure of this angle. 

63. Sidereal Time. The intervals between the successive 
transits of any fixed point of the sphere (as, for instance, of 
a star which has no proper motion) over the same meridian 
would be perfectly equal, were it not for the variable effect 
of nutation, (Hersch. Ast., Art. 327.) This correction, 
arising from a change in the position of the earth's axis, is 
most perceptible in its effect upon the transit of stars near 
the vanishing point of that axis, i. e. near the poles of the 
heavens. Hence, for the exact measurement of time, we use 
the transits of some point of the equator, as the vernal 
equinox. This point is often called the first point of Aries. 
Its usual symbol is c p. 

64. The interval between two successive transits of the 
vernal equinox is a sidereal day ; and such a day is regarded 
as commencing at the instant of the transit of that point. 



TIME. 73 

The sidereal time is then O h O m s . This instant is sometimes 
called sidereal noon. 

The effect of nutation and precession in changing the time 
of the transit of the vernal equinox is so nearly the same at 
'two successive transits, that the sidereal days thus defined 
[are sensibly equal. It is unnecessary, then, except in refined 
discussions, to discriminate between mean and apparent 
sidereal time. 

65. The sidereal time at any instant is the hour-angle of 
the vernal equinox at that instant, and is reckoned on the 
equator from the meridian westward around the entire cir- 
cle, that is, from to 24 h . It is equal to the right ascension 
of the meridian at the same instant. 

66. Solar Time. The interval between two successive 
transits of the sun over a given meridian is a solar day, and 
the hour-angle of the sun at any instant is the solar time of 
that instant. 

In consequence of the motion of the earth about the sun 
from west to east, the sun appears to have a like motion 
among the stars at such a rate that it increases its right as- 
cension daily nearly 1°, or 4 m of time. With reference to 
the fixed stars, it therefore arrives at the meridian each day 
about 4 m later than on the previous day ; consequently, solar 
days are about 4 m longer than sidereal days. 

67. Apparent and Mean Solar Time. If the sun changed 
its right ascension uniformly each day, solar days would be 
exactly equal. But the sun's motion in right ascension is 
not uniform, varying from 3 m 35 s to 4 m 26 s in a solar day. 
There are two reasons for this, — 

1st. The sun does not move in the equator, but in the 
ecliptic. 

2d. Its motion in the ecliptic is not uniform, being most 
rapid at the time of the earth's perihelion, about January 1, 
and slowest at the time of the aphelion, about July 2. 



74 NAVIGATION. 

To obtain a uniform measure of time depending on the 
sun's motion, the following method is adopted. A fictitious 
sun, called a mean sun, is supposed to move uniformly in the 
ecliptic at such a rate as to return to the perigee and apogee 
at the same time with the true sun. A second mean sun is 
also supposed to move uniformly in the equator at the same 
rate that the first moves in the ecliptic, and to return to each 
equinox at the same time with the first mean sun. 

The time which is measured by the motion of this second 
mean sun is uniform in its increase, and is called mean time. 

That which is denoted by the true sun is called true or 
apparent time. 

The difference between mean and apparent time is called 
the equation of time. It is also the difference of the right 
ascensions of the true and mean suns. 

The instant of transit of the true sun over a given meridian 
is called apparent noon. The instant of transit of the second 
mean sun is called mean noon. The mean time is then h 
m s . 

Mean noon occurs, then, sometimes before and sometimes 
after apparent noon, the greatest difference being about 16 m , 
early in November. 

68. Astronomical Time. The solar day (apparent or 
mean) is regarded by astronomers as commencing at noon 
(apparent or mean), and is divided into 24 hours, numbered 
successively from to 24. 

Astronomical time (apparent or mean) is, then, the hour- 
angle of the sun (true or mean) reckoned on the equator 
westward throughout the entire circle from h to 24 h . 

69. Civil Time. For the common purposes of life, it is 
more convenient to begin the day at midnight, that is, when 
the sun is on the meridian below the horizon, or at the sun's 
lower transit. The civil day begins 12 h before the astro- 
nomical day of the same date; and is divided into two 



TIME. 75 

periods of 12 h each, namely, from midnight to noon, marked 
A.M. (ante-meridian), and from noon to midnight, marked 
P.M. (post-meridian). Both apparent and mean time are 
used. 

The affixes A.M. and P.M. distinguish civil time from as- 
tronomical time. During the P.M. period, this is the only 
distinction, — the day, hours, &c. being the same in both. 

70. Sea-Time. Formerly, in sea-usage, the day was sup- 
posed to commence at noon, 12 h before the civil day, and 24 h 
before the astronomical day of the same date ; and was di- 
vided into two periods, the same as the civil day. Sea-time 
is now rarely used. 

71. To convert civil into astronomical time, it is only 
necessary to drop the A.M. or P.M., and when the civil time 
is A.M., deduct l d from the day and increase the hours by 
12b. 

To convert astronomical into civil time, if the hours are 
less than 12 h , simply affix P.M. ; if the hours are 12 h or more 
than 12 h , deduct 12 h , add l d , and affix A.M. 

Examples. 

Ast. Time. Civil Time. 

dhme dhms 

1860 May 10 14 15 10 = 1860 May 11 2 15 10 A.M. 

1862 Sept. 8 9 19 20 = 1862 Sept. 8 9 19 20 P.M. 

1863 Jan. 3 23 22 16 = 1863 Jan. 4 11 22 16 A.M. 
1863 Jan. 4 3 30 = 1863 Jan. 4 3 30 P.M. 

72. The hour-angle of the sun (true or mean), at any me- 
ridian, is called the local (apparent or mean) solar time. The 
hour-angle of the sun (true or mean) at Greenwich at the 
same instant is the corresponding Greemcich time. 

So also the hour-angle of °p at any meridian, and its hour- 
angle at Greenwich at the same instant, are corresponding 
local and Greenwich sidereal times. 




76 NAVIGATION. 

73. Tlxe difference of the local times of any two meridians 
is equal to the difference of longitude of those meridians. 

Demonstration. In Fig. 20, let 
P M, P M' be the celestial merid- 
ians of two places ; 
P S, the declination circle through 9^ 

the sun (true or mean) ; 
M P S, the hour-angle of the sun 

at all places whose meridian is 

P M, will be the local time (apparent or mean) at those 

places ; so also 
M' P S will be the corresponding local time at all places 

whose meridian is P M 7 ; and 
M P M'= MPS- W P S will be the difference of longitude 

of the two meridians. 

If P °p is the equinoctial colure, 
M P °p and M' P °p will be the corresponding sidereal times 

at the two meridians ; still, however, 
M P M'= MPf-MTT. 

The proposition is true, then, whether the times compared 
are apparent, mean, or sidereal. 

The difference of longitude is here expressed in time. It 
is readily reduced to arc by observing that 

24*= 360° 1 rio_ 4m 

; . \ or <^ 1' = 4 s 

IB =: 15' J l "" 15 

Iu comparing corresponding times of different meridians, 
the most easterly meridian is that at which the time is 
greatest. 

74. If (Fig. 20) PM is the meridian of Greenwich, 

M P S is the Greenwich solar time, and 

M P M' the longitude of the meridian P M'. 



TIME. 77 

MPM'=MPS-M'PS; 
so also MPM'=MPf-M'Pfj 

or, the longitude of any meridian is equal to the difference 
between the local time of that meridian and the correspond- 
ihg Greenicich time. 

75. If we put 

r o =MPS, the Greenwich time, 

T = M' P S, the corresponding local time, 

A = MP M', the longitude of the meridian, P M', 

we have X = T — T, ) ,» . 

and T =T+^ J V ; 

in which A is + for west longitudes, and T and T are sup- 
posed to be reckoned always westward from their respective 
meridians from h to 24 h ; that is, T and T are the astro- 
nomical times, which should always be used in all astro- 
nomical computations. 

76. Usually the first operation in most computations of 
nautical astronomy is to convert the local civil time into 
the corresponding astronomical time (Art. 71). 

The Greenwich time should never be otherwise expressed 
than astronomically. On this account it would be conven- 
ient to have chronometers intended for nautical or astronomi- 
cal purposes marked from h to 24 h , instead of h to 12 h as 
is now customary with sea-chronometers. 
* 77. The second operation often required is to convert the 
local astronomical time into Greenwich time. For this we 
have (72), which numerically is 

( + when the longitude is west) 
• ~~ \ — when it is east, 

and, in words, gives the following 

Rule. Having expressed the local time astronomically, 
add. the longitude, if west ; subtract it, if east: the result is 
the corresponding Greenwich time. 



78 NAVIGATION. 

TIME. 

Examples. 

1. In Long. 76° 32' W., the local time being 1861, April 
Id 9 h 3 m 10 s A.M., what is the Greenwich time ? 

Local Ast. T. == March 31 d 21 h 3 m 10 3 

Longitude — +568 

G. T. = April "1 2 9 18 

2. In Long. 30° E., the loc. time being March 20 d 6 h 3 m 
A.M., what is the G. T. ? 

Loc. Ast. T. = March 19 d 18 h 3 m 
Long. = — 2 

G. T. s= March 19 16 3 

3. In Long. 105° 15' E., the loc. time being Aug. 21 d 4 h 3 m 
P.M., what is the G. T. ? 

Loc. Ast. T. = Aug. 21 d 4 h 3 m 
Long. = — 7 1 

G. T. == Aug. 20 21 2 

4. Long. 175° 30' W., Loc. T. Sept. 30 d 8 h 10 m A.M. 

G. T. Sept. 30 d 7 h 52 m . 

5. Long. 165° 0' E., Loc. T. Feb. l d 7 h ll m P.M. 

G. T. Jan. 31 d 20 h ll m . 

6. Long. 72° 30' W., Loc. T. April 10 d 7 h 10 ra A.M. 

G. T. 10 d h m . 

7. Long. 100° 30' E., Loc. T. June l d b m A.M. (or mid- 
night.) G. T. May 31 d 5 h 18 m . 

8.- Long. 75° W., Loc. T. June 3 d h m M. (noon.) 
G. T. June 3 d 5 h m . 

By reversing this process, that is by subtracting the longi- 
tude if west, or adding it if east, we may reduce the Green- 
wich time to the corresponding local time. 

78. When observations are noted by a chronometer regu- 
lated to Greenwich time, an approximate knowledge of the 



TIME. 79 

longitude and local time is necessary in order to determine 
whether the chronometer time is A.M. or P.M., and thus fix 
the true Greenwich date. If the time is A.M., the hours 
must be increased by 12\ 

Examples. 

1. In Long. 5 h W., about 3 h P.M., on Aug. 3 d , the Green- 
wich chronometer shows 8 h ll ra 7% and is fast of G. T. 6 ra 10 9 . 
What is the G. T. ? 

Approx. Loc. T. Aug. 3 d 3 h G. Chro. SMl m T 

Long. + 5 Correction — 6 10 

Approx. G. T. Aug. 3 d 8 h G. T. Aug. 3 d 8 b 4 m 57 s 

2. In Long. 10 h E., about l h A.M., on Dec. 7 d , the G. Chro. 
shows 3 b 14 m 13 8 .5, and is fast 25 m 18 9 .7, find the G. T. 

Approx. Loc. T. Dec. 6 d 13 h G. Chro. 3 h 14 m 13 3 .5 

Long. — 10 Correction — 25 18 9 .7 

Approx. G. T. Dec. 6 d 3 h G. T. Dec. 6 d 2 h 48 m 54 s .8 

3. In Long. 9 h 12 m W., about 2 h A.M., on Feb. 13 d , the 
G. Chro. shows ll h 27 m 13 9 .3, and is fast 30 ra 30 9 .3, find the 
G. T. 

Approx. Loc. T.Feb. 12 d 14 h m G. Chro. ll h 27 ra 13 9 .3 

Long. + 9 12 Correction — 30 30 9 .3 

Approx. G. T. Feb. 12 d 23 h 12 m G. T. Feb. 12 d 22 h 56 m 43\0 

The operations on the approximate times may be per- 
formed mentally. 



CHAPTER IV. 

THE NAUTICAL ALMANAC. 

79. The American Ephemeris and Nautical Almanac " is 
divided into two distinct parts. One part is designed for 
the special use of navigators, and is adapted to the meridian 
of Greenwich. The other is suited to the convenience of 
astronomers, on this continent particularly, and is adapted 
to the meridian of Washington." 

80. The Nautical part of this Ephemeris and the British 
Nautical Almanac give at regular intervals of Greenwich 
time the apparent right ascensions and declinations of the 
sun, moon, planets, and principal fixed stars, the equation of 
time, the horizontal parallaxes and semidiameters of the sun, 
moon, and planets, and other quantities, some of which little 
concern the navigator, but are needed by astronomers. 

81. Before we can find the value of any of these quanti- 
ties for a given local time, we must first find the correspond- 
ing Greenwich time (Art. 77). When this time is exactly 
one of the instants for which the required quantity is put 
down in the Almanac, it is only necessary to transcribe the 
quantity as it is there given. When, as is mostly the case, 
the time falls between two Almanac dates, the required 
quantity is to be obtained by interpolation. And generally, 
except when great precision is desired, it is sufficient to use 
first differences only ; that is, regard the changes of the 
quantity as proportional to the small intervals of time, 
which are employed. 



THE NAUTICAL ALMANAC. 81 

Thus, for a day the change of the sun's right ascension 
may be regarded as uniform, so that for l h it is ^\ of the 
daily change ; for 2 h , ^ T ; and in general for any part of a 
day it will be the same part of the daily change. 

Generally, then, if 
A represent the quantity in the Almanac, for a date pre- 
ceding the given Greenwich time ; 
J x , its change in the time T ; 

£, the time after the Almanac date for which the value of 
the quantity is required, expressed in the same unit as T ; 
and 
A, the required value ; 
we have, 

A = A,+ ^A lm (73) 

When A is increasing, A l has the same sign as A ; but 
when A is decreasing, A x has the opposite sign. 

82. If the given time is nearer the subsequent than the 
preceding Almanac date, it may be convenient to interpolate 
backward. If, then, A x represent the quantity in the Al- 
manac for a subsequent Greenwich date, and t' the time be- 
fore the Almanac date, we have 

A^A-^A,. (74) 

83. The Almanac contains the rate of change, or difference 
of each of the principal quantities for some unit of time. 
Thus, in the Ephemeris of the sun and planets the "Diff. 
for l h ," in part of that of the moon, the " Diff. for l m ," are 
given. If t or t r is expressed in the same unit of time as 
that for which the " Diff.," J 19 is given, formulas (73) and 
(74) become 

Thus, for using hourly differences, we w r ish the hours, 



82 NAVIGATION. 

minutes, <fcc, of the Greenwich time expressed in hours and 
parts of an hour ; for using the differences for l m , we wish 
the minutes and seconds of Greenwich time expressed in 
minutes and parts of a minute. Decimal parts are usually 
most convenient; though some computers prefer aliquot 
parts. 

84. The quantities in the Almanac, as commonly in other 
mathematical tables, are approximate numbers, that is, each 
is given only to the nearest unit of the lowest retained or- 
der ; and no refinement of interpolation can give a result to 
a higher degree of precision. In interpolating, more than 
one lower order in any case is superfluous. Thus, the sun's 
declination is given to the nearest 0".l, and in no way can 
we by interpolation obtain a value which will be reliable 
within a narrower limit. 

Moreover, the Greenwich times are uncertain to a greater 
or less extent; and if first differences only are used, the in- 
terpolated result can be regarded as true only within much 
wider limits than the approximation of the Ephemeris. 

In interpolating, then, it is well to consider the degree of 
approximation which is wanted in any particular case ; and 
if the nearest 1', or 10", or 1" suffices, contract the interpo- 
lation so as to retain at the most one lower order ; or else, 
consider the degree of approximation attainable in any par- 
ticular case, and contract the work so as to retain only the 
reliable figures. All lower orders are superfluous, and are 
deceptive, as giving the appearance of a higher degree of 
accuracy than has actually been obtained ; as, for instance, 
using tenths and hundredths of seconds, when the data will 
give a result reliable within 2' or 3' only. 

A convenient method of contracting the multiplication 
and division of decimals is given in a pamphlet on the sub- 
ject. 

85. Should it be desirable to interpolate more accurately 



THE NAUTICAL ALMANAC. 83 

than can be done by first differences alone, the reduction for 
second differences may be introduced by a simple process. 

Let A 2 he the change of A l in the time T\ then instead 
of J 15 as found in the Almanac for the nearest Greenwich 
date, we may substitute 

4+ ^V 4; (76) 

that is, the value of A u interpolated for -J £, or to the middle 
instant between the Almanac date and the given time. 
This is simply using the mean rate of change during the in- 
terval. 

If A v is a "Diff. for l h " given for the Almanac for each 
day, T = 24 b ; if A, is a "Diff. for l ra " given in the Al- 
manac for each hour, T' = 60 ra . 

The interpolation of A x to the middle instant may often be 
performed mentally. 

Example. If the sun's right ascension for 1865, Jan. 30, 
8 h 9 m time be required, we find in the Almanac, 

for Jan. 30 h A v — 10".246 

4 = — 0*.035 

31 h A x = 10.211 

and by interpolation for Jan. 30 4 h , the middle instant be- 
tween Jan. 30 h and Jan. 30 8 h , 

4 = 10".246 — 0".006 == 10".240, 
which is the mean hourly change in the interval from b 
to 8 h . 

86. Formula (76), however, applies only to the American 
Ephemeris, where the differences for l h or for l m , which are 
designated by J x , are given for the same instants of Green- 
wich time as the functions, A, to which they belong.* 

In the British Almanac they are given for the middle in- 
stant between two dates.f For instance, the "Diff. for l h " 

* The "Prop. Logs, of Diff." of the Lunar Distances are given for the 
middle instant. 

f In the Ephemeris of the Planets it is otherwise. 



84 NAVIGATION. 

given in each as if for noon Jan. l d , is in the American 
Ephemeris the change per hour at Jan. l d b ; in the British 
Almanac, the change per hour, Jan. l d 12 h , or midnight. 
For the British Almanac (76) becomes 

A x being taken from the same line or for the same date as 
A . This is the date preceding that of A x . 

87. Problem 21. To find from the Almanac a required 
quantity for a given mean time at a given place. 

Solutio7i. The preceding considerations le#d to the follow- 
ing rule : — 

1. Express the given mean time astronomically, stating 
the day as well as the hours, &c, and reduce it to Green- 
wich mean time by adding the longitude if west, subtracting 
if east. 

2. Take from the Almanac for the nearest preceding mean 
time date the required quantity and the corresponding 
"Diff. for l h ," or "Diff. for l m ," noting the name or sign of 
each; multiply the "Diff. for l h " by the hours and parts of 
an hour, or the " Diff. for l m " by the minutes and parts of 
a minute, of the remaining Greenwich time ; and add the 
product algebraically. 

Or, take out for the nearest subsequent date the required 
quantity and its difference;* multiply the "Diff." by the 
hours and parts of an hour, or the minutes and parts of a 
minute, of the interval from the given Greenwich date to 
the Almanac date ; and subtract the product algebraically. 

When greater precision is required, interpolate the differ- 
ence to the middle instant between the given Greenwich 
date and the Almanac date, and use the result instead of 
the difference given in the Almanac. 

* From the British Almanac, the difference given for the preceding date 
should be taken. 



THE NAUTICAL ALIUlSAG. 85 

This rule is applicable to all those quantities, which are 
given at regular intervals of Greenwich mean time, except 
the moon's meridian passage and age and lunar distances. 

For the " Sidereal Time at Greenwich Mean Noon," on 
p. II. of each month, the "Diff. for l h " is 9 S .856 ; the part 
of Tab. II. of the American Ephemeris, for converting a 
mean solar into a sidereal i?iterval, may be used for the in- 
terpolation. 

The " Mean Time of Sidereal h ," on p. III., is given at 
intervals of 24 h of sidereal time. The "Diff. for l h " is 
— 9 s . 830 ; and the part of Tab. II. for converting a sidereal 
into a mean solar interval may be used. 

88. The quantities given in the American Ephemeris for 
Washington mean time may be interpolated in the same 
way, by reducing the local time to Washington time instead 
of to Greenwich time. 

89. The apparent places of the fixed stars are given in 
the British Almanac for the upper transit over the meridian 
of Greenwich ; in the American, for the upper transit over 
the meridian of Washington. In the latter, the Washington 
mean time is given. The sidereal time at either place for 
the instant of transit is the right ascension of the star (Art. 
65). 

Generally, the position given for the nearest day suffices. 
But if greater precision is required, it is necessary to reduce 
the local mean time to the sidereal time of the prime meri- 
dian, and interpolate for it. 

90. In the following examples, the required quantities are 
taken from the American Ephemeris, and interpolated to 
the nearest second by 1st differences (75), and to the highest 
precision attainable by 2d differences (76). 



86 NAVIGATION. 

Examples. 
For the local mean time, 1865, Jan 30 d 9 b 14 m 30 s A.M. in 
Long. 163° 14' W., find the following quantities from the 
Nautical Almanac : — 

The equation of time. 

O's right ascension, D's right ascension, 

O's declination, S's declination, 

O's semidiameter, D's horizontal parallax, 

O's horizontal parallax ; J)'s semidiameter ; 
Jupiter's right ascension, declination, horizontal parallax, 
and semidiameter ; 

The R. ascension and declination of a Scorpii (Antares). 
Ast. mean time, 1865, Jan. 29 d 21 h 14 m 30 s 
Long. +10 52 56 

G. mean time, 1865, Jan. 30 8 7 26 

8 7.433 
8.1239 



1. Tlie Equation of Time. (Page II.) 




hmss h ms s 

Jan. 30 13 38.8 +0.390 in 1 13 38.81 +0.384 (at 4 h ) 


j 3.1 in 8 
+ 3.2 ( .1 .124 + 3.12 ^ 

13 42 13 41.93 


r 3.072 in 8 h 
38 .1 
8 .02 
2 .004 


ibtractive from mean time. 




2. The O's right ascension. (Page II.) 




hhms s hms, 

Jan. 30 20 52 34.8 +10.246 20 52 34.81 


+ 10.240 (at 4 h ) 


( 82.0 in S h 
+ 1 23.3 ] 1.0 .1 + 1 23.19 
I .3 .024 
20 53 58 2.0 53 58.00 


f 81.920 in 8 h 
1.024 .1 
.205 .02 
[ 41 .004 


3. The O's declination. (Page II.) 




Jan. 30 — 1°7 33 56.7 +41.44 —17 33 56.7 


+ 41.57 (at 4 h ) 


f331.5 in S h 
+ 5 36.6 J 4.1 .1 +5 37.7 
1 .8 .02 
— 17 28 20 [ .2 .004 —17 28 19.0 


f 332.56 in 8 h 
J 4.16 .1 
1 .83 .02 
I .17 .004 



THE NAUTICAL ALMANAC. 87 

4. The O's semidiameter (p. I.) and hor. parallax (p. 250). 

O'smeanH. P. 8.5116 

for Jan. 30 8J71 

5. 7%6 B's right ascension. (Page XII.) 

hhms 8 hms h 

Jan. 30 8 23 50 28.8 +2,3127 23 50 28.77 +2.3127 (at 3 m .7) 



h 

Jan. 30 


16 16.22 —0.14 in l d 
— .05 in 8 h = i d 




16 16.17 



( 15.2 
+ 16.2 -| .9 

( -I 
23 50 45 


in7 m 
.4 
.033 


f 15.189 in 7 m 
+ 16.19 J .925 .4 
1 69 .03 
23 50 44.96 [ 1 .003 


6. The D's declination. 


(Page 


XII.) 


Jan. 30 8 +2 7 48.5 +12.178 




+ 2 7 48.5 +12.177 (at 3"\7) 


( 85.2 
+ 1 30.5 \ 4.9 

( -4 

+ 2 9 19 


in 7 m 
.04 
.033 


( 85.24 in 7 m 
+ 1 30.5 \ 4.87 .04 
( .41 .033 
+ 2 9 19.0 



7. The D's horizontal parallax. (Page IV.) 

Jan. 30 60 25.7 -084 60 25.7 — 093 (at 4 h ) 

„ Q ( 6~7~ in 8 h h A j 1AA in 8* 

- 6 ' 8 j .1 .12 - M j .11 .12 

60 19 60 18.1 



8. The D's semidiameter. (Page IV.) 

Jan. 30 16 29.7 — 5^5 in l d 16 29.7 

— 1.8 in 8^= i d —2.1 = — 7".55 x .2729 

16 28 16 27.6 

111 Art. (59) we have for the moon, s = .2729 n ; whence 

As = .2729 An: 
so that the reduction of the semidiameter may be readily 
found by multiplying that of the horizontal parallax by .2729, 
as in the above example. This coefficient admits of a con- 
venient set of aliquot parts ; for .2730 = .25 + .025 — .002, 
so that A s '= (i + T V — jf o) A n nearly. 



88 NAVIGATION". 


9. 2£'s right ascension. (Page 


230.) 


Jan. 30 17 22 24.4 +1.974 


h m s a 

17 22 24.40 +1.971 (at 4 h ) 


j 15.8 in 8 h 
+ 16.0 I .2 .12 

17 22 40 


f 15.768 in 8 h 
+ 16.01 J .197 .1 
1 39 .02 
17 22 40.41 [ 8 .004 


10. 71' s declination. (Page 230.) 


Jan. 30 — 2°2 39 32.4 —1.85 


O 1 II II 

— 22 39 32.4 -1.84 


( 14.8 in 8 h 
— 15.0 \ .2 .1 

— 22 39 47 


( 14.72 in 8 h 
— 14.9 \ .18 .1 
( 4 .02 
— 22 39 47.3 



11. 2['s semidiameter and horizontal parallax. (Page 
384.) 

Jan. 30, Vert. sem. diam., 16".29, Hor. Par., 1".46. 

This is the vertical semidiameter when the planet is on the meridian, or 
the semidiameter in the direction of the declination circle of the planet. 
The polar, or minor, semidiameter of the elliptic disk is given on page 230. 

12. The right ascension and declination of a Scorpii. 
(Antares.) 

The Washington mean time is Jan. 30 2 h 59 m , or Jan. 
30.12. On page 258, which serves as an index, the mean 
R. A. is 16 h 21 m . The apparent R. A. and Dec. are for Jan. 
30.8 m. t. Washington, 

R. A. 16 h 21™ 8 S .67 + 9 .33 Dec— 26° 7' 37".l — 0*8. 
Change in — d .7 — .02 + .1 

16 21 8.65 ~- 26 7 37.0 

91. Problem 22. To find from the Almanac the sun's 
right ascension and declination, and the equation of time 
for a given apparent time at a given place. 

Solution. Tins differs from the preceding problem simply 
in using the apparent instead of the mean time, and in tak- 
ing the quantities from page I. for the month, where they 
are given for apparent noon, instead of from page II., where 
they are given for mean noon. 



THE NAUTICAL ALMANAC. 89 

Examples. 

Find the Q's R. A. and Dec. and the equation of time 
for 1865 Jan. 30 9 h m 48 9 A.M. apparent time in Long. 
163° 14' W. 

Ast. app. time 1865 Jan. 29 21 h m 48 8 



Long. 


+ 10 52 56 




G. app. time 


29 7 53 44 
7 53.73 
7.896 




h m s s 

0's R. A. 20 52 37.1+10.246 


O / 11 

q's dec— 17 33 47.2 


+ 41.44 


+ 120.9 |W-J 

20 53 58 


+ 5 27.2 
— 17 28 20 


f 290.1 
J 33.1 
1 3.7 

I -3 


m s s 

Equation of time + 13 38.9 + 0.390 






+ 3.1 
+ 13 42.0 







92. Problem 23. To find the right ascension and decli- 
nation of the sun, and the equation of time at apparent 
noon of a given place, or when the sun is on the meridian. 

Solution. The local apparent time is h m s . The Green- 
wich apparent time is then equal to the longitude if west, 
that is, it is after the noon of the same date by a number of 
hours, &c, equal to the longitude. If the longitude is east, 
the Greenwich apparent time is before the noon of the same 
date by a number of hours, &c, equal to the longitude. 

Hence, take these quantities from the Almanac for Green- 
wich apparent noon (page I.) of the same day as the local 
(civil) day, and apply a correction equal to the hourly differ- 
ence multiplied by the hours and parts of an hour of the 
longitude ; observing to add or subtract the correction, ac- 
cording as the numbers in the Almanac may require, for a 
time after noon, if the longitude is west; for a time before 
noon, if the longitude is east. 



90 NAVIGATION. 

The hourly differences from the British Almanac should 
be taken as given for the preceding instead of the same day 
in east longitude. 

Examples. 
1. Find the sun's right ascension and declination, and the 
equation of time for apparent noon, 1865, Jan. 30, in Long. 
163° 14' W. 

h m s h m b s 

Long. + 10 52 56 O's R. A. 20 32 37.14+ 10.238 

f 102.38 in 10 h 
I 5.129 30 m 
+ 151.36^ 3.413 20 

.427 2 30 s 
20 34 28.50 I 7 25 



O'sdec— 17 33 47.2 +41.62 Eq. of T. + 13 38.90 + 0.382 

+ 7 32.6 



'416.2 T3.82 in 10* 

20.81 +4.15 J .191 30™ 

h 13.87 J .127 20 

— 17 26 14.6 | 1.73 +-13 43.05 [ .016 2| 



I 
2. For apparent noon, 1865, March 21, in Long. 163° 14' E. 

h m 8 h m s g 

Long. —10 52 56 0's R. A. —0 3 20.43 +9.099 

-10 52.93 f9 ^r 

—10.882 — 1 39.02 | ^79 

1 41.41 f ' 18 



0's dec. +0 21 44.9 +- 59.21 Eq. of T. +7 15.59 —0.757 

T592.1 
— 10 44.3 J 47.37 +-8.24 

| 4.74 
+ 11 0.6 [ .12 

3. For apparent noon, 1 865, March 20, in Long. 150° 35' W. 

h m b h m s s 

Long. +- 10 2 20 O's R. A. 23 59 42.07+ 9.101 

91.01 in 10 h 
+ 1 31.36 .303 2 ra 

51 20* 

1 13.43 




THE KAUTTCAL ALMANAC. 



91 



O's dec. —0 1 56.2 + 59.23 
+ 5 54.6 
+ 3 58.4 



592.3 in 10 h 
1.97 2 m 

.33 20 9 



Eq. of T. + 7 33.72—0.755 

— 7. 



+ 7 26.14 



75.5 
25 
4 



In the 1st and 2d examples, the Diffs. for l h have been in- 
terpolated for 5\5 or half the longitude, forward in the first, 
back in the second : in the third they have been interpolated 
forward for 5\ 

93. Problem 24. To find the right ascension of the mean 
sun for a given time and place. 

Solution. At the instant of mean noon, or when the mean 
sun is on the meridian, at any place, the right ascension of 
the mean sun is equal to the sidereal time. The quantity on 
page II. of each month, in the Almanac, called " sidereal 
time," is also the right ascension of the mean sun at Green- 
wich mean noon, and may be interpolated for a given local 
time in the same way as the right ascension of the true sun. 
(Prob. 21.) The constant "Diff. for l h " is 9 S .856. A table 
for converting mean time into sidereal time intervals (Tab. 
II.) facilitates the interpolation. 

We have also the right ascension of the mean sun equal 
to that of the true sun + the equation of time, using for 
the equation of time the sign of its application to mean 
time. 

94. Problem 25. To find the mean time of the moonh 
transit over a given meridian on a given day. 

Solution. The Almanac contains the mean time of each 
transit of the moon over the meridian of Greenwich (page 
IV.). This mean time is the hour-angle of the mean sun 
(Art. 72) when the moon is on the meridian; and is there- 
fore the difference of right ascension of the moon and the 
mean sun. As this difference is constantly increasing, in 
consequence of the moon's more rapid increase of right 



92 NAVIGATION. 

ascension, the mean time of each transit is later than that 
of the one preceding by a number of minutes, varying, ac- 
cording to the rate of the moon's motion, from 40 m to 66 m . 

If, then, T x and T 2 denote the mean times of two succes- 
sive transits of the moon over the Greenwich meridian, 
T 2 — T x is the retardation of the moon in passing over 24 h 
of longitude ; so that for any longitude X (expressed in 
hours) the retardation is nearly 

tiW-Ti). (78) 

The mean time of a transit is, then, reduced from the 
Greenwich to any other meridian by interpolating for the 
longitude ; forward, if the longitude is west; backward, if 
the longitude is east, since east longitudes are regarded as 
negative. 

The American Ephemeris gives also the hourly differences, 
which facilitate the interpolation. For greater exactness, 
these differences may be interpolated for half the longitude. 
The practical rule will be : — 

Take from the Almanac the mean time of meridian passage 
for the given astronomical* day, and add to it the product 
of the " Diff. for l h " by the longitude in hours, if the longi- 
tude is west; subtract that product if the longitude is east. 

From the British Almanac the daily retardation may be 
found by taking the difference for two successive transits ; 
and the reduction by multiplying it by the longitude in parts 
of a day ; or it may be taken from Tab. XXVIII. (Bowd.) 
The mean time of meridian passage for the given day, and 
that for the day folloicing in west longitude, or for the day 
preceding in east longitude, are those which are commonly 



* It is important to notice whether the mean time of transit is more or 
less than 12 h . In the former case, the astronomical day is l d less than the 
civil day. 



THE NAUTICAL ALMANAC. 93 

used. (Bowd., p. 170.) But it is more exact to use half 
the difference of the times of meridian passage for the day 
preceding and the day following the given day : 2V °f tn ^ s i s 
the " Diff. for l h " of the American Ephemeris. 

The times of transit are given only to tenths of a minute, 
which suffices the purposes of the navigator. They may be 
found more exactly for any meridian by the method here- 
after given in Prob. 33. 

95. Problem 26. To find on a given day the mean time 
of transit of a planet over a given meridian. 

Solution. The mean time of each meridian passage at 
Greenwich is given, in the Almanac, for each planet. It 
may be reduced to any meridian in the same way as for the 
moon ; except that, in the case of an acceleration^ the sign 
of the reduction is reversed. 

Examples. 

1. In Long. 100° 15' W., find the times of meridian pass- 
age of the moon and Jupiter for 1865, June 6 (civil day). 

Long. + 6 h 41 m s = 6\683. 

h m s h m m 

M. T. of mer. pass. June 6 9 59.6 + L99 June 5 12 41.8 —^45 in l d 

f 1L94 ( LlT in 6 h 

+ 13.3 J 1.19 — 1.2 \ .11 .6 
1 .16 (1 .08 

June 6 10 12.9 [ 1 June 5 12 40.6 

or June 6 40.6 A.M. 

2. In Long. 100° 15' E. for 1865, June 6 (civil day), find 
the times of meridian passage of the moon and Jupiter. 

Long. — 6 h 41 m 9 = — 6 h .683. 

J> U 

M. T. of mer. pass. June 6 9 59.6 +1^ June 5 12 41.8 —4.45 in l d 

T1L82 (T.Tlin6 h 

— 13.2 J 1.18 + 1.1 



( 1.11 in 



] .16 (1 .08 

June 6 9 46.4 [ 1 June 5 12 43.0 

or June 6 43.0 A.M. 



94 NAVIGATION. 

In the case of the moon the hourly differences have been 
interpolated for half the longitude. 

96. Problem 27. To find the right ascension or declines 
tion of the moon, or a planet, at the time of its transit over 
a given meridian on a given day. 

Solution. Find the local mean time of transit, as in Pro- 
blem 25 ; deduce the corresponding Greenwich time by ap- 
plying the longitude ; and for this Greenwich time take out 
the right ascension or declination, as in Problem 21. 

If the time of transit has been noted by a clock or chrono- 
meter, regulated to either local or Greenwich time, it should 
be used in preference to the time of transit computed from 
the Almanac. 

97. Problem 28. To find the Greenwich mean time of a 
given lunar distance. 

Solution. The angular distances of the moon from the sun, 
the principal planets, and several selected stars, are given in 
the Almanac for each 3 h of Greenwich mean time. 
If d represent the given distance ; 

c? , the nearest distance of the same body in the Almanac 

preceding in time the given distance ; 
J 15 the change of distance in 3 h ; 
t, the required time (in hours) from the date of d ; 
by (75) we have approximately, using 1st differences only, 

whence, for the inverse interpolation, 

* = !(<?-<£), (79) 

or, with t in seconds of time, which is better for computation, 

t = ——(d-d ), (80) 

in which it is most convenient to express A x and (d — d ) in 
seconds. 



THE NAUTICAL ALMANAC. 95 

Then by logarithms : 

-rx i 10800 fMS 

log t = log (d - do) '+ log—,-, (81) 



Ji 



— ^- is the change of distance in 1" ; hence log — j- is the 

ar. complement of the " log diff. for I s ." 

It is given in the Almanac for the middle instant between 
the tabulated distances under the head " P. L.* of Diff." ; 
the index, which is 0, and the separatrix being omitted. 

In the same way, if 
d Y represent the distance in the Almanac following the given 

distance ; and 
t\ the interval before the date of c? 19 

we shall have by (76 

and t f =-j(d l —d)^ 

or with t' in seconds, and by logarithms, 

r i / t 7x t 10800 , . 

log t = log (d- d) +log-^-. (82) 

The computation is simplified by using a table of "loga- 
rithms of small arcs in space or time."f It differs from the 
common table of logarithms only in having the argument in 
sexagesimal instead of natural numbers. With such a table 
it is unnecessary to reduce differences of distance to seconds, 
or to first find the intervals of time in seconds. 

From (81) and (82) we have the following rule : Find in 
the Almanac the two distances between which the given 
distance falls ; take out the nearest of these, the hours of 
Greenwich time over it and the " P. L. of Diff." between 

* Proportional Logarithm. 

f Tab. I. of the American Ephemeris before 1865 : Tab. IX. of Chauvenet's 
Lunar Method. 



96 NAVIGATION. 

them. Find the difference between the distance taken from 
the Almanac and the given distance ; and to the log. of 
this difference add the " P. L. of Diff." from the Almanac ; 
the sum is the log. of an interval of time to be added to the 
hours of Greenwich time taken from the Almanac, when the 
earlier Almanac distance is used ; to be subtracted from the 
hours of Greenwich time when the later Almanac distance is 
used. (Chauvenet's Lunar Method, p. 8.) 

98. The result, however, may not be sufficiently approxi- 
mate, owing to the neglect of 2d differences. To correct it 
for 2d differences, Tab. X. of Chauvenet's Method, Tab. IT. 
of the Almanac, or the table on p. 245 of Bowditch, may be 
used. For either, take the difference between the two Prop. 
Logs., which precede and follow the one taken from the Al- 
manac. With half this difference, and the interval of time 
just found, enter the table and take out the seconds, which 
are to be added to the approximate Greenwich time when 
the Prop. Logs, are decreasing ', but subtracted when they are 
increasing. 

2d differences may also be introduced by first finding, or 
estimating, the Greenwich mean time to the nearest 10 m , and 
interpolating the Prop. Log. in the Almanac to the middle 
instant between that time and the Almanac hour used, as in 
Art. 88 for direct interpolation. 

99. Maskelyne, the author of the present arrangement of 
Lunar distances, to facilitate their interpolation, devised 
what he chose to call proportional logarithms. 

If n represent any number of seconds, either of space or 

time, the proportional logarithm of n is the log of . 

Tab. XXII. (Bowd.) contains these proportional loga- 
rithms for each second of n from to 3°, or to 3 h , the argu- 
ment being in ° ' " or in h m fl . But such a table is less use- 
ful for other purposes than Tab. I. of the American Ephe- 
meris, previously refer red to. 



THE NAUTICAL ALMANAC. 97 

Dividing both members of (80) by 10800, and inverting, 
we have 

10800 _ A 10800 

t ~ 10800 X cl—dj 
and, 

P. log t = P. log {d-d Q )-V. log 4 U (83) 

which accords with the rule on page 231. (Bowd.) 

100. Examples. 

1865, Oct. 31, the distance of Fomalhaut from the moon's 
centre is 42° 3' 35", what is the Greenwich mean time ? 



Oct, 31 15 h d — 41 17 58 P. log 0.3142 % cliff. — 91 

d — d Q — 45 37 log 3.4373 

t = + l h 34 ra 3 s log 3.7515 

Red. for 2d diff. + 28 
G. m. time Oct, 3116 34 31 

or, by back interpolation, 

* c?=42° 3' 35" 
Oct, 31 18 h ^.= 42 45 17 P. log 0.3142 £ diff. — 91 
d x — d — 41 42 log 3.3983 

t r = — l h 25 m 58 9 log 3.7125 

Red. for 2d diff. + 28 
G. m. time Oct. 31 16 34 30 

The P. L. interpolated to 15 h 47 m is 0.3163, and to I7 h I7 ra 
is 0.3118. Had these been used instead of 0.3142, the result- 
ing values of t and t' would have included the reduction for 
2d difference. 



CHAPTER V. 

CONVERSION OF THE SEVERAL KINDS OF TIMR- 
RELATION OF TIME AND HOUR-ANGLES. 

CONVERSION OF TIME 

101. Problem 29. To convert apparent into mean time, 
or mean into apparent time. 

Solution. For the same instant, let 
T m represent the local mean time ; 
T a , the local apparent time ; and 

JEj the equation of time with the sign of its application to 
apparent time. 

Then, since the equation of time is the difference of mean 
and apparent times (Art. 67), 

T m =T a +EA 

T a = T m -E\ I 84 ' 

The reduction, then, is made by finding from the Almanac 
the equation of time for a given apparent time, from page 
I. of the month (Prob. 22), or for a given mean time from 
page II. (Prob. 21), and applying it to the given time 
according to the precept at the head of the column where it 
is found. 

102. The equation of time on page I. is sometimes called the 
mean time of apparent noon ; and on page II. the apparent 
time of mean noon. Regarding it, as in (84), as the reduc- 
tion of apparent to mean time, it indicates, when additive 



CONVERSION" OF TIME. 99 

and increasing, or subtractive and decreasing, that mean 
time is gaming on apparent time. 

103. Problem 30. To convert a mean into a sidereal time 
interval, or a sidereal into a mean time interval. 

Solution. The sidereal year is 365.25636 mean solar days, 
or 366.25636 sidereal days; so that the same interval of 
time which is measured by 365 d . 25636 reckoned in mean 
time, is measured by 366 d .25636 if reckoned in sidereal time 
(Herscb., Ast. 305). Since both are uniform measures of 
time, if we represent any interval by 

t, if expressed in mean time, 

s, if expressed in sidereal time, then 

■= 1.0027379; 



365.25636 



wdience 



s = 1.0027379 t = £ + .0027379 t, (85) 

t == 0.9972696 s = s— .0027304 s, (86) 

by which the reduction from one to the other may be made. 

The computation is facilitated by Tab. II. of the American 
Ephemeris, the first part of which, for converting sidereal 
into mean solar time, contains for each minute of s the value 
of .0027304 s ; the second part, for converting mean solar 
into sidereal time, contains for each minute of t the value of 
.0027379 t. 

Tables LI. and LII. (Bowd.) contain the same quantities 
to tenths of seconds only. 

104. If in (86) t = 24 h ; *= 24 h 3 m 56 s .5553 ; ormnmean 
solar day sidereal time gains on mean time 3 ra 56 s .5553.» In 
l h of mean time the gain is 9 s . 8565. 

If in (87) s = 24 h ; t = 24 h — 3 m 55 9 .9094 ; or in a sidereal 
day mean time loses on sidereal time 3 m 5 5 s . 90 94. In l h of 
sidereal time the loss is 9 8 .8296. 

If t and s in the last term are expressed in hours, (85) and 
(86) become 



100 NAVIGATION, 

s = t + 9\85Q5 % 



t — s— 9 9 .8296 s; J ' y 

by which the reductions may be more readily calculated, 
when the tables are not at hand. 

105. Problem 31. To convert mean time at a given place 
into sidereal time. 

Solution. Let 

X represent the longitude of the place, expressed in time, 

+ when ivest, 
T, the local mean time, 
$, the corresponding sidereal time, 
t, the interval from mean noon in mean time (differing from 

^only by omitting the day), 
/S Y , the same interval in sidereal time, 
/Sq, the sidereal time of mean noon at Greenwich, 
/S Y ' , the sidereal time of mean noon at the place ; 
then, since X expresses the Greenwich time of local noon, 
(Art. 92), 

S' =S + .0027379 A; 1 
evidently S = s + S' . L (88) 

and by (86) s == t +.0027379 t; J 

whence we have 

S=t +£ + .0027379 (X + t). (89) 

The Almanac (page II.) contains JS for each Greenwich 
mean noon, under the head " Sidereal Time." It should be 
taken out for the given astronomical day of the place ; 
.0027379 X is then the reduction for longitude, additive in 
west longitude, subtractive in east. It, as well as .0027379 £, 
the reduction to a sidereal interval, may be taken from the 
second part of Tab. II. of the Almanac, or from Tab. LI. 
(Bowd.) ; or either may be computed by (86) or first of (87). 

From (89), then, we have the following rule : 

To the local mean time add the sidereal time of Green- 



CONVERSION OF TIME. 101 

toich mean noon of the given astronomical day, the reduction 
of this sidereal time for the longitude of the place, and the 
reduction of the hours, minutes, &c, of the mean time to a 
sidereal interval. 

The astronomical (solar) day is usually retained. But if 
it be desirable to state the sidereal day, as well as the hours, 
&c, of the sidereal time, we prefix to S the sidereal day at 
the instant of mean noon, which is the same as the astro- 
nomical day after the vernal equinox of each year ; one day 
less before that date. At the instant of the vernal equinox 
the sidereal time and mean solar time coincide. Before that 
time the mean sun transits before the vernal equinox ; after 
that time it transits after the vernal equinox. 

106. T+ X is the Greenwich mean time. When this is 
given, or found in the course of computation, it will be more 
convenient to take out S for the Greenwich day, and the 
combined reduction, .0027379 (t + X), for the hours, minutes, 
&c, of Greenwich mean time, instead of for t and X sepa- 
rately. 

It should be noted, however, that in the first method 
(Art. 105), S Q is taken out for the local day ; in this, it is 
taken out for the Greenwich day, provided X + 1, as used, 
expresses properly the Greenwich time. 

107. S + .0027379 (t + X) is the "sidereal time" of the 
Almanac interpolated for the Greenwich mean time. It is 
more convenient to term it the right ascension of the mean 
sun (Art. 93) ; and then the translation of (89) will be, the 
sidereal time is equal to the right ascension of the mean. sun 
+ the mean time. 

This is also evident from Fig. 21, in which 




102 NAVIGATION. 

P is the pole ; 
P M, the meridian ; 
f, the vernal equinox ; 
T M, the equator. 
°p M is also the right ascension 
of the meridian, and measures 
MPf, the hour-angle of T, or 

the sidereal time (Art. 65). 

If P S is the declination-circle passing through the mean sun, 
T S is the right ascension of the mean sun, and 
MPS is its hour-angle or the mean time (Art. 72), and is 

measured by the arc of the equator, S M. 
Evidently T M = T S + S M. (90) 

The hour-angles MPf, MPS, are reckoned from the 
meridian toward the west ; hour-angles east from the me- 
ridian are then regarded as negative. 

If P S is the declination-circle of the true sun, then will 

T S be the right ascension, and 

MPS the hour-angle of the true sun ; and 

S M will measure the apparent time, 

and the interpretation of (90) will be, the sidereal time is 

equal to the right ascension of the true sun + the apparent 

time. 

Examples. 

1. Find the sidereal time of 1865, Jan. 30, 10 11 15 m 26 9 .6, 
ast. mean time in long. 150° 13' 10" (10 h m 52 s .7) W. 
First Method. Second Method. 

h m s h m ■ s 



L. m. t. Jan. 30 10 15 26.6 


L. m. t. Jan. 30 10 15 26.6 


S 20 38 56.00 


Long. +10 52.7 


Red. for long. + 1 38.71 


G. m. t. Jan. 30 20 16 19.3 


Red. ofL.m.t. + 141.10 


L. m. t. 10 15 26.6 


Sid. t. 6 57 42.4 


S 20 38 56.00 




Red. for G. m. t, + 3 19.81 




Sid. t. 6 57 42.4 



CONVERSION OF TIME. 103 

2. Find the sidereal time of 1865, Jan. 30, 10 h 15 ra 26 8 .6, 
ast. mean time in long. 10 h m 5 2 s . 7 E. 

h m s 

L. m. t. Jan. 30 10 15 26.6 



S 20 36 56.00 

Red. for long. — 1 38.71 

Red. of L. m. t. + 

Sid. t. 6 54 25.0 



1 38.71 ) •■ „ mi _ 

-i 4-1 10 f 2d part of lab. II. 



3. Find the sidereal time of 1865, Sept. 25, 21 h 16 m 15% 
in long. 60° 13' (=4 h m 52 s ) W. 

h m s h m s 

L. m. t. Sept. 25 21 16 15 L. m. t. Sept, 25 21 16 15 
S 12 17 15.9 Long. + 4 52 

G. m. t, Sept. 26 1 17 7 



Red. for long. 


+ 39.6 


G. 


Red. of L. m. t. 


+ 3 29.7 


S 


Sid. t. 


* 9 37 40 


R( 



12 21 12.7 
Red. for G. m. t. + 12.6 

Sid. t. 9 37 40 



4. Find the sidereal time of 1865, Sept. 25, 3 h 16 m 15 9 .0, 
in long. 8 h 16 m 25 s .3 E. 

o 

h m 8 h ra s 

L. m. t. Sept. 25 3 16 15.0 L. m. t. Sept. 25 3 16 15.0 

S 12 17 15.89 Long. — 8 16 25^3 

Red. for long. — 1 21.55 G. m. t. Sept, 24 18 59 49.7 



Red. for L. m. t. + 32,24 S 12 13 19.34 

Sid. t. 15 32 41.6 Red. for G. m. t. + 3 7.25 

Sid. t, 15 32 41.6 

108. Problem 32. To convert sidereal time at any place 
into mean time, 

1st Solution. The sidereal time at mean noon at the place 
is from (88) 

g' =z S + . 0027379 A; ■ 

the sidereal interval from mean noon, 

s = S—S f = S—S — .0027379 A ; (91) 

and from (86) the corresponding mean time interval, 

£ = s--.0027304 s. (92) 



104 NAVIGATION". 

The mean time T is completed by prefixing to t the astro- 
nomical day. 

From (91) and (92) we have the following rule: 

From the local sidereal time subtract the sidereal time of 
Greenwich 'mean noon of the given astroyiomical day and 
the reduction of this sidereal time for the longitude of the 
place ; and from the sidereal interval thus obtained subtract 
the reduction to a mean time interval ; and to the residt pre- 
fix the giv n astronomical day. 

The local sidereal time may be increased by 24 h if neces- 
sary. The reduction for longitude, .0027379 A, may be taken 
from the 2d part of Tab. II. of the Almanac, or from Tab. 
LI. (Bowd.) ; numerically, it is subtr active ixiicest longitude, 
additive in east, as applied to the given sidereal time. The 
reduction of the sidereal interval, .00273.04 s, may be taken 
from the 1st part of Tab. II., or from Tab. LII. (Bowd.), 
and is always subtractive. 

2d Solution. Let 

M represent the "mean time of the preceding sidereal h " * 

at Greenwich ; 
M ' , the " mean time of the preceding sidereal h " at the 

place ; 
S, the interval from h in sidereal time ; 
£, the same interval in mean time : 

then, since X will be the sidereal interval between the 
Greenwich and local sidereal h (Art. 92), 

M\ — M — . 0027304 A, 
evidently, T— t + M' , 

and by (86) t =: £—.0027304 IS ; 

whence we have 

T= S+3I— .0027304 (X + jS). (93) 

* It is equal to 24 h — the right ascension of the mean sun. In the British 
Almanac it is called " Mean time of transit of first point of Aries." 



CONVERSION OF TIME. 105 

The Almanac (page III.) contains Jf for the Greenwich 
sidereal h on each mean day. The Almanac date of the 
preceding sidereal h is generally the same as the local 
astronomical date when the sidereal time is less than the 
"sidereal time at mean noon" (page II.), but l d less when 
the sidereal time is greater than that at mean noon. The 
doubtful case is w r hen the mean time is within 4 m of noon : 
the comparison must then be made with the sidereal time at 
the nearest local mean noon. 

The reduction of M to the local meridian is — .0027304 A, 
which may be taken from the 1st part of Tab. II., or from 
Tab. LIL (Bowd.) It is subtr active in icest longitude, ad- 
ditive in east. 

The reduction of the sidereal interval, .0027304 /S", maybe 
taken from the same tables ; it is always subtractive. 

The combined reduction, .00^304 (A-f £), may be taken 
out for the Greenwich sidereal time, (A + $), instead of for 
A and S separately ; but with these precautions, that when 
A + $>24 l1 , M may be taken out for l d later than stated in 
the previous precept, and interpolated for the excess of 
(A + /S) over 24 h ; and when (A + £) is negative, to retain its 
negative character, or else take out M Q for one day earlier. 

3d Solution. From (89) we have 

t = £-[£ + .0027379 (£ + A)], (94) 

so that, when the Greenwich mean time (£+A) is sufficiently 
known, we may find for it the right ascension of the mean 
sun, (Art. 107) 

$, + .0027379 (£ + A), 

and subtract it from the given sidereal time : or, the mean 
time is equal to the sidereal time — the right ascension of the 
mean sun. So also we have from Art. 107 the precept:— 
the apparent time is equal to the sidereal time— the right 
ascension of the true sun. 



106 NAVIGATION. 



Examples. 

1. 1865, Jan. 30 (ast. day), in long 10 h m 52 9 .7 TV\, the 
sidereal time is 6 h 57 m 42. 8 4 ; find the mean time. 

h m s h m s 

L. sid. t. 6 57 42.4 L. sid. t. 6 57 42.4 

— # (Jan. 30) —20 38 56.00 M (Jan. 30) 3 20 31.06 

— Red. for long. — 1 38.71 Red. for long. — 1 38.44 
Sid. int. 10 17 7.69 Red. of sid. t. — 1 8.43 
Red. of sid. int. — 1 41.10 L. m. t. Jan. 30 10 15 26 .6 
L. m. t. Jan. 30 10 15 26.6 



2. 1865, Jan. 30, (ast. day,) in long. 10 h m 52 s .7 E. ? the 
sidereal time is 6 h 54 m 25\0; what is the mean time? 

h m • s h m s 

L. sid. t. 6 54 25^) L. sid. t. 6 54 25.0 

— # (Jan. 30) —20 38 56.00 M (Jan 30) 3 20 31.06 

— Red. for long. + 1 38.71 Red. for long. + 1 38.44 
Sid. int. 10 17 7.71 Red. of sid. t. — 1 7.89 
Red. of sid. int. — 1 41.10 L. m. t. Jan. 30 10 15 26.6 
L. m. t. Jan. 30 10 15 26.6 

3. 1865, Sept. 26, 9 b A.M., in long. 4 h m 52 9 W., the sidereal 
time is 9 h 37 m 40 s . 1 ; find the mean time. 

h m s h m s 

L. sid. t. 9 37 40.1 L. sid. t. 9 37 40.1 

— S (Sept. 25) —12 17 15.89 M (Sept. 25) 11 40 48.98 

— Red. for long. — 39.57 Red. for long. — 39.46 
Sid. int. 21 19 44.64 Red. of sid. t. —1 34.64 
Red. of sid. int. — 3 29.65 L. m. t. Sept. 25 21 16 15.0 
L. m. t. Sept. 25 21 16 15.0 

4. 1865, Sept. 25, 3 h P.M., in long. 8 h 16 m 25 s .3 E.,the side- 
real time is 15 h 32 m 41\6 ; find the mean time. 



RELATION OF HOUR-ANGLES AND TIME. 



107 



L. sid. t. 

— S (Sept. 25) - 

— Reel, for long. 
Sid. int. 

Red. of sid. int. 



15 32 41.6 
12 17 15.89 

+ 1 21.55 
3 16 47.26 

— 32.24 



L. m. t. Sept. 25 3 16 15.0 



L. sid. t. 
M (Sept. 24) 
Red. for long. 
Red. of sid. t. 



15 32 41.6 
11 44 44.89 
+ 1 21.33 
— 2 32.80 



L. m. t. Sept. 25 3 16 15.0 



RELATION OF HOUR-ANGLES AND TIME. 

109. Problem 33. To find the mean time of meridian 
transit of a celestial body, the longitude of the place or the 
Greenvrich time being known. 

Solution. In the case of the sun the instant of meridian 
transit is apparent noon of the place ; for which we have 
(84) 

T m — E, the equation of time, 

which can he taken from page I. of the Almanac, and inter- 
polated for the longitude, which in this case is also the 
Greenwich apparent time ; or from page II., and interpo- 
lated for the Greenwich mean time. When E is subtractive, 
the subtraction from the number of days can be performed. 

The apparent right ascension of any body at the instant 
of its meridian transit is also the right ascension of the me- 
ridian, or sidereal time. (Art. 65.) It suffices therefore to 
find the right ascension of the body, and, regarding it as the 
sidereal time, reduce it to mean time by Problem 31. 

The American Ephemeris contains the apparent right as- 
censions of two hundred principal stars for the upper cul- 
minations at Washington ; the British Almanac contains the 
positions of one hundred for the upper culminations at 
Greenwich. They are reduced to any other meridian, when 
necessary, by interpolating for the longitude. 

The right ascensions of the moon are given for each hour, 
and of the planets for each noon, of Greenwich mean time, 



108 NAVIGATION. 

and may be found for a given Greenwich mean time by 
Problem 21. If, however, the longitude of the place is given, 
the local mean time of transit of the moon, or a planet, may 
first be found from the Almanac to the nearest minute or 
tenth (Probs. 25, 26) ; then for this mean time the right as- 
censions of the moon, or of the planet (Prob. 21), and of 
the mean sun (Prob. 24), may be computed. Subtracting 
the right ascension of the mean sun from the right ascen- 
sion of the moon, or planet, will give the mean time of 
transit (Prob. 32, 3d solution). If it differ sensibly from 
that previously obtained, the process may be repeated with 
this new approximation. 

If the time of transit has been noted by a clock, or chro- 
nometer, regulated either to local or Greenwich time, it 
should be used in preference to the approximate time of 
transit found from the Almanac in computing the right 
ascensions. 

The American Ephemeris contains also the right ascen- 
sions of the moon and principal planets at their transits of 
the upper meridian at Washington. They can be reduced 
to any other meridian by interpolating for the longitude 
from Washington. 

This solution will give the time of the upper culmination 
of a heavenly body. To find the time of a lower culmi- 
nation, 12 b may be added to the right ascension of the body, 
if sufficiently Avell known ; or, as is generally preferable, 
12 h may be added to the longitude of the place. The in- 
stant of a lower culmination on any meridian will be that 
of an upper culmination on the opposite meridian. 

Examples. 

1. Find the times of meridian passage of the moon and 
Jupiter for 1865, June 6 (civil day), in long 100° 15' W. 
(Example 1, Art. 95, p. 93.) 



RELATION OF HOUR-ANGLES AND TIME. 109 



Approx. m. t. 

Long. 

G. m. t. 


2) 

h m 

June 6 10 12.9 

+ 6 41.0 

June 6 16 53.9 




June 
June 


U 

h m 

5 12 40.6 

+ 6 41.0 

5 19 21.6 




R. Asn. 

Red. for G. m. t. 

R. As'n at transit 


h m s 

15 13 28.54-j- 
+ 1 53.66 | 

15 15 22.20 


s 

2.1088 
105.440 
6.326 

1.898 




h m s 

17 39 52.S0- 
— 25.64 h 

17 39 27.16 


s 

- 1.340 

"13.40 

11.760 

.402 

80 


So 

Red. for G. m. t. 


4 59 3S.32 
+ 2 46.56 

5 2 24.88 






4 55 41.76 
+ 3 10.82 
4 58 52.58 




M. t. of transit, 


June 6 10 12 57.32 
t. + 8.32 




June 5 


12 40 34.58 




Diflf. from approx. 


— 1.42 





- „, O0 ( Ch. of R. A. + .117 

In o,o2- _ CLof ^ _ <0()9 

M. t. of transit, June 6 10 12 57.43 

110. Problem 34. To find the hour-angle of the sun for 
a given place and time. 

Solution. The hour-angle of the sun, reckoned from the 
upper meridian toward the west, is the apparent time 
reckoned astronomically (Art. 72). Its hour-angle east of 
the meridian is negative, and numerically equal to 24 h — the 
apparent time. 

A given mean or sidereal time must then be converted 
into apparent time ; for this, the longitude, or the Green- 
wich time, must be known approximately. 

111. Problem 35. To find the hour-angle of the moon, a 
planet, or a fixed star, for a given p>lace and time. 

Solution. In Fig. 21, as described in Art. 104, 

T M is the right ascension of the meridian, and measures 
M P Y, the sidereal time. 

Let 




110 NAVIGATION. 

PS be the declination-circle of 

the mean sun, then 
°f S is the right ascension of the 

mean sun, and cfr 

M P S is the mean time, and is 

measured by the arc of the 

equator, S M. 

Let 

P M' be the declination-circle of some other celestial body ; 

then 
Y M' is its right ascension, and 
M P M' is its hour-angle, and is measured by the arc M' M. 

From the figure, 

M' M =TM -Y M'= Y S + SM-Y M'. (95) 

If Y S is the right ascension of the true sun, 
S M will measure the apparent time. 

From (95), then, we have the following rule : — 

To a given apparent time add the right ascension of the 
true sun ; or to a given mean time add the right ascension 
of the mean sun, to find the corresponding sidereal time. 
Then from the sidereal time subtract the body's right ascen- 
sion ; the difference is the hour-angle west from the meri- 
dian. If it is more than 12 h , it maybe subtracted from 24 h : 
the hour-angle, then, is — , or east of the meridian. It is 
necessary to know the longitude, or the Greenwich time, 
sufficiently near to find the right ascensions of the sun and 
body. 

112. Problem 3G. To find the local time, given the hour- 
angle of the sun and the Greenwich time. 

Solution. The hour-angle reckoned westward is itself the 
local apparent time, which may be reduced to mean or side- 
real time (Probs. 29, 30), as may be required. The Green- 



RELATION OF HOUR- ANGLES AND TIME. Ill 

wich time, or the longitude of the place, is needed only for 
this reduction. 

113. Pkoblem 37. To find the local time, given the hour- 
angle of some celestial body and the Greenwich time. 

Solution. Find from the Almanac for the Greenwich time 
(Prob. 21) the right ascension of the body. Then, from (95), 
we have 

¥M=YM'+M'M 3 

from which, and Arts. 105, 107, we have the following rule, 
regarding hour-angles to the east as negative : — 

To the right ascension of the body add its hour-angle, the 
result is the sidereal time. From this subtracting the right 
ascension of the true sun gives the apparent time ; or the 
right ascension of the mean sun gives the mean time. 

The Greenwich time is needed for finding the required 
right ascensions. 

If the longitude of the place is given, but not the Green- 
wich time, we may first use an estimated Greenwich time, 
and then revise the computations with a corrected value, 
until the assumed and computed values sufficiently agree. 

Examples. 

1. 1865, Jan. 16, 12 h 15 m 17 S .6, mean time in long. 150° 
13' 10" W., find the hour-angle of the moon. 

h m s h m s 

L. m. t. Jan. 16 12 15 17.6 L. m. t. Jan. 16 12 15 17.6 

Long. + 10 52.7 S 19 43 44.22 

G. m. t. Jan. 16 22 16 10.3 Red, for long. + 1 38.71 

^sR.A.tJan. 16 22 lj ) 1148 31.61 + 1 9 .8584 Red. of L. m. t. + 2 0.79 

fl8 .584 
Red. for G. m. t. + 30.05 J 11 .150 L. sid. t. 8 2 41.32 

1 .186 

t .133 

D's R, A. at date 11 49 1.66 

D's hour-angle — 3 46 20.34 



112 NAVIGATION. 

2. 1865, Jan. 16 22 h 16 m 10\3, G. mean time, the moon's 
hour- angle is — 3 b 46 m 20 9 .3 ; find the local mean time. 

h m s 

D's hour-angle — 3 46 20.3 

j)'s R. A. (Jan. 16 22 h ) +11 48 31.61 + K8584 

f 18 .584~ 
Red. for G. m. t. + 30.05 J 11 .150 

] .186 
L. sid. t. 8 2 41.36 I .133 

—S (Jan. 16) -19 43 44.22 

—Red. for G. m. t. — - 3 39.50 
L. m. t. Jan. 16 12 15 17.6 

Subtracting this from the G. m.. t. gives for the longitude 
10 h m 52 b #7 ^y 

3. 1865, Jan. 16, 12 h (nearly) in long. 150° 13' 10" W., 
the moon's hour-angle is — -3 h 46 m 20 s .3 ; find the local mean, 
time. 

h m s h m m 

Long. 10 52.7 D's mer. pass. Jan. 16 15 50.7 +1.74 

2>'s h. ang.— 3 h 46 m .3 Red. for long. +17.4 

. gh - g ( ch. of R. A. - 7 .0 Jan. 16 16 8.1 

' ( -ch. of JS Q + .6 - 3 52.7 

1st approx. L. m. t. Jan. 16 12 15.4 
Long. +10 0.8 

1st approx. G. m. t. Jan. 16 22 16.2 



D's h. ang. 


h m a 

— 3 46 20.3 








D's R. A. (Jan. 16 


22 h ) + ll 48 31.61 


+ l s 
(18 


.8584 
.584 




Red. for G. m. t. 


+ 30.11- 


i 11 


.150 ch. in — 1 8 .6 
.372 


—.046 


L. sid. t. 


8 2 41.42 








— S (Jan. 16) 


— 19 43 44.22 








—Red. for G. m. t. 


— 3 39.50 




— ch. in — 1 8 .6 


+ .004 


2d L. m. t. Jan. 16 12 15 17.70 




cor. for —I s . 6 


—.04 



Long. 10 52.7 

2d G. m. t. Jan. 22 22 16 10.4 

Diff. from 1st G. m. t. —1.6 



3dL. m. t. Jan. 16 12 15 17.7 



CHAPTER VI. 

NAUTICAL ASTRONOMY. 

ALTITUDES. AZIMUTHS. HOUR-ANGLES AND TIME. 

115. Nautical Astronomy comprises those problems of 
Spherical Astronomy which are used in determining geo- 
graphical positions, or in finding the corrections of the in- 
struments employed. In general, they admit of a much 
more refined application on shore, where more delicate and 
stable instruments can be used, than is possible at sea, 
where the instability of the waves and the uncertainty of 
the sea-horizon present practical obstacles, both to precision 
in observations and to the accuracy of the results, which can- 
not be obviated. 

116. In the problems which are here discussed the follow- 
ing notation will be employed : — 

L z= the latitude of the place of observation ; 
h = the true altitude of a celestial body ; 
z = 90°— A, its zenith distance ; 
d = its declination ; 
^ = its polar distance ; 
t =: its hour-angle ; 
Z = its azimuth. 
Let the diagram (Fig. 22) represent the projection of the 
celestial sphere on the plane of the horizon of a place : — 

Z, the zenith of the place ; 

N Z S, its meridian ; 



114 



NAVIGATION. 




P, the elevated pole, or that 
whose name is the same as 
that of the latitude ; 

M, the position of a celestial 
body ; 

Z M H, a vertical circle ; and 
P M, a declination-circle, through 
M. 

Then, in the spherical triangle 

Fig. 22. P M Z, 

PZ = 90°— X, the co-latitude of the place ; 

P M = p — 90°— d, the polar distance of M ; 

ZM= 90°— A, the complement of its altitude, or its zenith 
distance ; 
Z P M = t, its hour-angle ; 
PZM = Z, its azimuth. 

The angle P M Z is rarely used, but is sometimes called 
the position angle of the body. 

This triangle, from its involving so many of the quantities 
which enter into astronomical problems, is called the astro- 
nomical triangle. As three of its parts are sufficient to de- 
termine the rest, if three of the five quantities X, d, A, £, and 
Z are known, the other two may be found by the usual 
formulas of spherical trigonometry. These admit, however, 
of modifications which better adapt them for practical use. 
The following articles point out how X, c?, A, and t may be 
obtained. 

117. The latitudes and longitudes of places on shore are 
given upon, charts, but more accurately in tables of geogra- 
phical positions, such as are found in books of sailing-direc- 
tions, and in Tab. LIV. (Bowd.) At sea it is sometimes 
necessary to assume them from the dead reckoning brought 
forward from preceding, or carried back from subsequent, 
determinations. (Bowd., p. 264.) 



NAUTICAL ASTRONOMY. 115 

118. The altitude of an object may be directly measured 
at sea above the sea-horizon with a quadrant or sextant ; on 
shore, with a sextant and artificial horizon, or with an alti- 
tude circle. All measurements with instruments require cor- 
rection for the errors of the instrument. Observed altitudes 
require reduction for refraction and parallax ; for semidia- 
meter, when a limb of the object is observed ; and at sea, 
for the dip of the horizon. The reductions for dip and re- 
fraction are subtract iv e ; for parallax, additive. Strictly, 
the reductions should be made in the following order : for 
instrumental errors, dip, refraction, parallax, semidiameter. 
In ordinary nautical practice it is unnecessary to observe 
this order. 

Following it we should have, — 

1st. The reading of the instrument with which an altitude 
is measured ; 

2d. The corrected reading or observed altitude of a limb ; 

3d. The apparent altitude of the limb ; 

4th. When corrected for refraction and parallax, the true 
altitude of the limb ; 

5th. The true altitude of the centre. 

Except with the sea-horizon, the observed and apparent 
altitudes are the same. For the fixed stars, and for the 
planets when their semidiameters are not taken into account, 
the altitudes of the limb and the centre are the same. 

Unless otherwise stated, the true altitude of the centre is 
the altitude which enters into the following problems, and is 
denoted by h. 

119. The hour-angle of a body can be found, when the 
local time and longitude, or the Greenwich time, are given. 
(Probs. 34, 35.) For noting the time of an observation, a 
clock, chronometer, or watch is used ; at sea, only the last 
two ; but it will be necessary to know how much it is too 
fast or too slow" of the particular time required. 






116 



NAVIGATION. 



120. The declination of a body can be found when the 
Greenwich time is known. (Prob. 21.) 

The polar distance of a heavenly body is the arc of the 
declination-circle between the body and the elevated pole 
of the place : that is, the north pole, when the place is in 
north latitude ; the south pole, when it is in south latitude. 
If 

PP' (Fig. 23) is the projec- 
tion of the declination-circle 
through an object, M ; 
P, the north pole ; 
P', the south pole ; 

E Q, the equator ; then the 
polar distances, 

P M = P Q-QM=90°-rf, 

P'M = P'Q + QM = 90°+ d. 

That is, the polar distance is 90°— d or 90° + J, according 
as the pole from which it is reckoned is N". or S. This, how- 
ever, is regarding declination, like the latitude, as positive 
when N., negative when S. 

To avoid, however, the double sign in the investigation 
of the formulas of Nautical Astronomy, we shall in most 
cases consider the declination, which is of the same name as 
the latitude, asfjositive, and that which is of a different name 
from the latitude, as negative ; hence the polar distance will 
be represented by 

jp=90°— d. 

When the declination is of a different name from the lati- 
tude, we have numerically 

p=90°+d. 




ALTITUDE AND AZIMUTH. 



117 



ALTITUDE AND AZIMUTH, 



121. Problem 38. To find the altitude and azimuth of a 
heavenly body at a given place and time. 

Solution. Find the decimation of the body and its hour- 
angle at the given time. (Probs. 21, 34, and 35.) 

Then in the spherical triangle PMZ (Fig- 24), we have 



given 






PZ = 


: 90°- 


-L, 


PM = 


90°- 


-d, 


ZPM = 


*, 




to find 






ZM = 


90°- 


-A, 


PZM = 


Z. 





By Sph.Trig. (122), (123), if in 
the triangle ABC (Fig. 25), we 
have given b, c, and A to find a 
and B, we have 

tan (p = tan b cos A, 

cos (c — <f) cos ~b 



cos a 



cos i 



. -^ sin (c — (f) cot A 

cot B == — v . J , 

sin $ 




which, by substituting the corresponding parts of the trian- 
gle P Z M, give 

tan 6 = cot d cos £, 



. , (sm (p + L) sin d 

sin h — - , 

cos (j) 

cotZ= cos( ^ +Z)coU , 

sm 



(96) 



J 



118 



NAVIGATION. 



If we put = 90°— </>', these become 
tan 0' == tan d sec t, 



sin h 



(cos (f) r — L) sin d 
sin </ 



, ~ sin (0 — Z) cot t 

COt Z = — -~ , 

COS 



(97) 



which afford the convenient precept, (j> r has the same name, 
or sign, as the declination, and is numerically in the same 
quadrant as t. 

122. When t = 6 h , 0'= 90°, and the 3d of (97) assumes 
an indeterminate form. But from the 1st we have 

tan d 



cot t = 



tan (J)' sin t ' 



which, substituted, gives 
cot Z = 



sin (0' — L) tan 6? 



(98) 



sin 0'' sin £ 
which may be used when t is near 6 h . 

123. A is the true altitude of M. If the apparent altitude 
is required, the parallax (Art. 54) must be subtracted, and 
the refraction (Art. 41) added. 

Z is the true bearing, or azimuth, of the body, reckoned 
from the N*. point of the horizon in north latitude, and from 
the S. point in south latitude. It is generally most conve- 
nient to reckon it as positive toward the east, which will re- 
quire in the above formulas — Z for Z, since t is positive when 
west. Restricting, however, Z numerically to 180°, it may 
be marked E. or W., like the hour-angle. 

121. In Fig. 24, if 31 m be drawn perpendicular to the 
meridian, then 
Pr/i = <p = 90° -0', 
Zm= (0 + Z)— 90°=Z— 0'; or, 
is the polar distance of m, 
0', its declination, 






ALTITUDE AND AZIMUTH. 119 

.£—</>, its zenith distance, positive, or of the same name as 
the latitude, toward the equator. A convenient precept is 
to mark it N. or S., according as the zenith is 3\T. or S. of 
the point m. 

m falls on the same side of the zenith as the equator when 
Z > 90° ; at the zenith when Z = 90° ; and on the 
same side as the elevated pole when Z < 90°. It falls 
between P and Z only when t and Z are both less than 
90°. 

125. In the case of a Ursse Minoris {Polaris), whose polar 
distance is 1° 25', the more convenient formulas derived 
from (96) will be, since p and are small, 

(p z=. p cos £, 

(which gives <f> within /; .5) 

sin h = sin (X + 0) — — , 
v YJ cos ^ 

~ tan » sin £ cos 6 

tan Z == -— -= — - ; 

cos (L + $) ' 

or approximately, 

h = X + 0, 

Z ' == p sin £ sec (X + </>) . 

Z is a maximum, or the star is at its greatest elongation, 
when the angle ZMP (Fig. 24), or Z n P (Fig. 30), is 90°. 
We then have 

sin Z = sin p sec X, 
or nearly 

Z ' = p sec X. 

126. Problem 39. To find the altitude of a heavenly body 
at a given place and time, when its azimuth is not required. 

Solution. The 1st and 2d of (96) or (97) maybe used; 
or, by Sph. Trig. (4), 

cos a = cos b cos c + sin b sin c cos A, 

we have sin h = sin L sin d+cos L cos d cos t; 

which, since cos t = 1 — 2 sin 2 -J- t, 




120 



NAVIGATION. 



cos (L—d) — 2 cos L cos d sin 2 it,) 

in t. \ 



(99) 



reduces to 

sin h 
or sin h = cos {L— d)— cos L cos c? versin 

(L—d) becomes numerically {L + d) when L and c? are 
of different names. 

Tab. XXVII. contains for the argument t in column P. M. 
the log sin % t in the column of sines ; which, doubled, is 
log sin 2 \t. It is well to note this, for mistakes are often 
made by regarding the logarithms in this table as log sin, 
log cos, &c, of t instead of ^ t. 

Tab. XXIII. contains for the argument t, log 2 sin 2 J t = 
log versin £, with the index increased by 5. 

It is sometimes necessary to compute the altitude of one, 
or both bodies, to use in connection with an observed lunar 
distance. The rules for this purpose on pp. 247, &c, Bowd., 
are derived from the above formulas. The result is evidently 
more accurate, the smaller the hour-angle £, especially if the 
altitude is near 90°. In these rules it is best to find the 
" sidereal time," or " right ascension of the meridian," from 
the mean local time, instead of the apparent (Art. 105). 

127. Problem: 40. To find the azimuth of a heavenly 
body from its observed altitude at a given place. 

Solution. In this the Greenwich time of the observation 
must be known sufficiently near for finding the declination of 

the body. The observed altitude 
must be reduced to the true alti- 
tude. Then in the triangle PZM 
we have given the three sides to 
find the angle P Z M. 

In the triangle ABC, putting 
s = $-(a + b+c), we have 

//sin s sin (s — b)\ 




COS 



B 



a- 



sin a sin c 



For the triangle P Z M, 



ALTITUDE AND AZIMUTH. 



121 



B = Z, a — 90° — h, A being the true altitude, 
b — p, the polar distance, 

c = 90°— X, the co-latitude, 
5= 90 o -i (i+A-p), 
5-6= 90° —J- (Z + A+p), 

and the formula becomes 



cos^Z— yt 



fcos \ (L + h+27) cos | (Z+^ — p) 
cos Z cos A 



-P) 



or, if we put s r = ^ (Z-f-A+i?), 

-. r7 //COS «' COS (V- 

cos -J- Z = 4/ =- , 

z r \ cos L cos A 

which accords with Bowditch's rule, p. 160. 
In a similar way we may find from the formula 

• i t-» // sin (s — a) sin (s — V)\ 

sin | B = |/ ( - 



(100) 



sin a sin c 



sin | 



in which 
or, if we put 



cos \ (co Z+A+d) sin \ (co L-\-Ji—cl) 
cos Z cos /*. 

CO L — 90°— Z; 



sm 



s*=-| (co Z + A-fd), 
^ ~_ /fcos s ff sin {s n —d) 



cos X cos A 



(101) 



(100) is preferred when Z> 90° ; (101), when Z< 90° 
If the body is in the visible horizon, then nearly 

h— — (33'+the dip). 
128. If the bearing of the body is observed with a com- 
pass at the same time that its altitude is measured, or if the 
bearing is observed and the local time noted, the decimation, 
or variation, of the compass can be found. For, the true 
azimuth, or bearing, of the body can be found from its alti- 
tude (Prob. 40), or from the local time (Prob. 38) ; and the 
magnetic declination is simply the difference of the true and 



122 



NAVIGATION. 



magnetic bearings of the same object, determined simulta- 
neously if the object is in motion. It is marked JE. when 
the true bearing is to the right of the magnetic bearing, 
W. when the true bearing is to the left of the magnetic 
bearing. (Bowd., p. 161.) 

129. The amplitude of a star when in the true horizon is 
its distance from the east or the west point, and is marked 
N. or S., according as it is north or south of that point. It 
is, therefore, the complement of the azimuth. 

Pkoblem 41. To find the amplitude of a heavenly body 
when in the horizon of a given place. 

Solution. Let the body be 
in the horizon at M (Fig. 27), 
A ==W M, its amplitude. The 
triangle P M N" is right angled 
at N, and there are given 



PN = Z, 

PM = 90°- 



■* 



to find 




S 

NM = Z=.90°-i. . Fig . 2T . 

We have cos P M = cos P N" cos 1ST M, 

or sin d = cos L cos Z, 

whence cos Z — sin A — sin d sec X, 



(102) 



as in Bowditch, p. 159. By (102) A is N or S like the de- 
clination. 

As the equat'or intersects the horizon of any place in the 
east or west points, it is plain that the star Avill rise and set 
north or south of these points, according as its declination is 
N. or S. 

Tab. VII. (Bowd.) contains the amplitude, A, for each 1° 
of latitude up to 60°, and each 1° of declination to 23°. The 
convenience of this table, in the case of the sun, is the only 



ALTITUDE AKD AZIMUTH. 



123 



reason for introducing amplitudes. It is generally best to 
express the bearing of an object by its azimuth. 

In this problem the body is supposed to be in the true 
horizon, or about (33' + the dip) above the visible horizon. 
Hence the rule to " observe the bearing of the sun, when its 
centre is about one of its diameters above the visible horizon." 
(Bowd., p. 158.) 



Examples. (Probs. 38—41.) 

1. 1865, Jan. 25, 2 h 33 m 13 s local mean time in lat. 49° 30' S. 
Ion o\ 102° 39' 15" E. ; required the sun's true altitude and 
azimuth. (97) 



h m s 

L. m. t. Jan. 25 2 33 13 (Jan. 25.) 0's dee, 
Lon£. 



Eq.oft, 



-6 50 37 18° 52' 48".7 S. - 37 ff .35 — 12 m 41 3 .67-0^561 
( 149 .4 (2^4 

G. m. t. Jan. 24 19 42 36 +2 40 .3 -J 1 .5 +2 .40 ) .11 

or Jan. 25 -4\39 18 55 29 S. ( 3 .4 - 12 39.3 ( 5 

Eq'noft. —12 39.3 

L. ap. t. 2 20 33.7 

*= 3o° 8' 25" 1. sec 0.08738 1. cot 0.15251 

d— 18 55 29 S. 1. tan 9.53512 1. sin 9.51098 
' <j>' — 22 44 50 S." 1. tan 9.62250 1. cosec 0.41266 1. sec 0.03516 
L— 49 30 



S. 

L—(p'= 26 45 10 X. 
h= 48 30 6 
Z=S 124 44 23 W. 



1. cos 9.95083 1. sin 9.65335 n 
1. sin 9.87447 

1. cot 9.84102 n 



\ 



The reduction for refraction and parallax of A=48°.5 is 
+ 45" ; and the apparent altitude is h f =z-iS° 30' 51". If the 
compass bearing of the sun at the same instant had been 
ST. 34° 20' W. = S. 145° 40' W., the magnetic declination 
would have been 20° 56' W. 

2. 1865, Sept. 27, 5 h 20 m 16 3 A.M. mean time in lat. 
50° 15' K, long. 87° 3.0' W.;' required the altitude and azi- 
muth of Venus. (97) 



124 



NAVIGATION. 



L. m. t. Sept. 26 17 20 16 Long. 



+ 5 50 



12 21 12.4 G. m. t. Sept. 26 23 10 16 =Sept. 27— 0\829 



Red. for long. 
Red. for L. m. t. 


+ 57.5 (Sept. 27) KA. 

h m s s 

+ 2 50.9 10 4 14.13 +11.645 


Dec. of Yenus. 

o / // // 

+ 12 32 25.0 —55.42 


L. sid. t. 


9.32 
5 45 16.8 —9.65 .23 


44.3 
+45.9 1.1 


R. A. of $ 


10 4 4.5 . . . -JW 


+ 12 33 11 - 5 


H. A. of $ 


—4 18 47.7 




*= 64 c 


41' 55" E. l.sec 0.36919 


1. cot 9.67461 


d = 12 


33 11 N. 1. tan 9.34766 1. sin 9.33714 


f = 27 


31 11 N. 1. tan 9.71685 1. cosec 0.33530 1. sec 0.05215 


L— 50 


15 N. 




f— L— 22 


43 49 S. 1. cos 9.96489 1. sin 9.58703 n 


h= 25 


42 42 1. sin 9.63733 


Z=N101 


38 17 E. 


1. cot 9.31379 n 



3. 1865, July 20, 5 h 58 m 20 s A.M., mean time in lat. 
38° 19' 20" N., long. 150° 15' 30" E. ; required the sun's azi- 
muth. (98) 



L. m. t. July 19 17 58 20 
Loner. —10 1 2 



(July 19) O's dec. Eq'n oft. 

o / // H m s 8 

+ 20 49 8.7 —27.60 —5 58.34 —0.165 











r 193.2 


{ 1.155 


G. 


m. t. 


July 19 7 57 18 = 7 h 


.955 -3 39.5^ 


24.8 
1.4 


-1.31 i .150 
( 9 


Ec 


l'n of t. —5 59.6 


+ 20 45 29 


I .1 


— 5 59 65 


L. 


ap. t. 


July 19 17 52 20.4 










t = 


91° 54' 54" E. 


1. cosee 1.47604 


n 


1. cosec 0.00024 




d-= 


20 45 29 N. 


1. tan 9.57868 




1. tan 9.57868 




¥ = 


95 2 17 N. 


1. tan 1.05472 


n 


1. cosec 0.00168 




£ = 


38 19 20 N. 






1. sin 9.92219 


*' 


— L = 


56 42 57 N. 
N 72 20 43 E. 






1. cot 9.50279 



4. Required the apparent altitudes of the sun and moon, 
Nov. 9, 1865, about 9 A.M., in lat. 18° 25' S., long. 84° 6' W. ; 
time by chro. 2 h 25 m 10 s ; chro. slow of G. m. t. 10 m 15 s . (99) 



ALTITUDE AND AZIMUTH. 



125 



T. by chro. 2 25 10 (Nov. 9) O's dec. Eq'n oft 

o / // ii m s s 

Chro. cor. +10 15 16 57 42.5 S. + 42.67 +16 0.76 — 0.230 

G. m. t Xov. 9 2 35 25 = 2 h .590 



— Long. —5 36 24 

L. m. t. Nov. 8 20 59 1 
Eq'noft. +16 0.2 

L. ap.t. Nov. 8 21 15 1.2 
L> 



d 



= 1° 25' 27" S. 



O's true alt.= 50 46 
Ref. and par. +40 

O's ap. alt. = 50 46 40 



+ 1 50.5 



^ = 16 59 33 S. 
Zr=18 25 S. 

t= — 2 h 44 m 58 s .8 



( 85.34 ( .460 

\ 21.33 — .60 \ .115 

( 3.84 +16 0.16 ( 2i 

1. cos 9.98061 

1. cos 9.97717 



cos .99969 

-.22511 

sin .77458 



1. shr 2 1 t 9.09360 



log 2 
log 



0.30103 
9.35241 



i* 



* 1. versin t = 9.39463 



L. m. t. Nov. 

/ied. for % 
Red of * m 
L. sid. t. 
D's R. A. 

Z-d = 



X)'s true alt. 
Par. and ref. 
j)'s app. alt. 



Par. and ref. 
D's app. alt. 



(Nov. 9 2 h ) tfsdec. 



D'sB.A. 



20 59 1 

15 10 44.2 13 4 38.9 N. —7.420 ! 42 28.70 + 2.128C 
+ 55.3 
+ 3 26.8 
12 14 7.3 
8 43 44.1 rf=13 16 
3 30 23.2 Z=18 25 



—4 22.8 




63.84 
+ 1 15.36*! 10.64 
8 43 44.1 



31° 25' 16" S. 

cos .85336 
—.36292 
29 22 10 sin .49044 

-49 Tab. XXIX. 



1. cos 9.98871 

1. cos 9.97717 

1. sin 2 i t 9.29290 

log 2 0.301 

loo; 9.55981 



90 U 
03 ) 



* 1. versin t = 9.59393 



28 33 (approx.) D's H. pa'x 56' 28".3 — 2".10 



-47 47 
28 34 23 



■ K K (4.2 

-°-° i l.s 



tt=56 22 .8 



log 3.5293 
1. sin 9.9437 
loo; 3.4730 



7t'rz:28 33' 

p=4& 32" 

ref. = — 145 
Par. and ref. =47 47 

By Tab. XIX. 47' 48" 



5. Find the altitude and azimuth of Polaris, 1865, Sept. 
25 8 h 15 m P.M., in lat. 49° 16' N"., long. 85° 16' W., (Art. 125). 



126 NAVIGATION. 



h m s 

L. m. t. 8 1.5 Long. 5 h 41™ 4 3 



S 12 IT 16 p = 124 27 log 3.7048 1. cos 9.99987 

Red. for long. + 56 t = 69 8 1. cos 9.5517 

Red. of L. m. t. + 121 <j, = 30 5 log 3.2565 1. sec 0.00002 

L. sid. t, 20 34 33 £+0 = 49 46 5 1. sec CU898 1. sin 9.88277 

R. A. of * 1 11 5 7i = 49 45 log p. 3.7048 1. sin h 9.88266 

H. ang. of * — 4 36 32 1. sin t. 9.9705 

Z=J$. 2 2 10 E. log 3.8651 

6. Required the greatest elongation of Polaris, 1865, 
Sept. 25, in lat 49° 16' NV 

p = 1 24 27 log 3.7048 or, 1. sin 8.3903 

L = 49 16 1. sec 0.1854 1. sec 0.1854 

Z = 2 9 26 log 3.8902 1. sin 8.5757 

% ' 

7. At sea, 1865, May 20, 15 h 23 m 16 s mean time Green- 
wich, in lat. 40° 15' S., long. 107° 15' W., the observed alti- 
tude of the sun's lower limb 10° 15' 20", index correction 
of sextant + 3' 20", height of eye 18 feet, bearing of sun 
by compass N. 41° 45' E. ; required the sun's azimuth and 
the magnetic declination or variation. (100.) 

G. m. t. May 20 15 h 23 m 16 s — 15 h .388 O's dec. 20° 2' 28".0 ]S T . + 30".79 

O 10° 15' 20" ( In. cor. + 3' 20" + 7 53.7 

+ 14 59 \ Dip. — 4 11 

(S. diam. + 15 50 20 10 22 N. 

7i = 10 30 19 1. sec 0.00734 

Z= 40 15 1. sec 0.11734 

p— 110 10 22 
2 s = 160 55 41 
s = 80 27 50 1. cos 9.21925 

s — p = — 29 42 32 1. cos 9.93880 

19.28273 

t Z = 64 148 1. cos 9.64137 

true Z - S. 128 3 36 E. = N. 51° 56' 24" E. 

Mag. N. 41 45 E. 

Variation 10 11 E. 




HOUR-ANGLE AND LOCAL TIME. 



121 



8. 1865, Sept. 20, in lat. 30° 25' N"., long. 50° 16' W., the 
compass bearing of the sun, when one of its diameters above 
the horizon, was S. 79° 30' W.; required its true bearing 
and the variation. (102.) 

L. ap. t. Sept. 20 6 h m (Tab.IX.) O'sdec 
Long. + 3 21 



0° 59' 32" N. — 58".4 



G. ap. t. Sept. 20 9 21 = 9\35 



— 9 6 



(Tab. TIL) ^ 



d = 
L = 

: TV. 0° 58' N*. true Z = N. 
mag. 
var. 



50 

30 .25 

S9 2 

]S T . 100 30 

11 28 



N. 
N. 
W. 
TV. 
TV. 



525. 6 
17. 5 

2. 9 

1. sin 8.1627 
1. sec 0.0643 
1. cos 8.2270 



9. On the same day at the same place, when the sun's 
centre was in the visible horizon, its compass bearing was 
S. 79° 30' W. ; height of eye 20 feet. 



p = 


89° 10 






z = 


30 25 


1. 


sec 0.0643 


h = — 


37 


1 


sec 


2s = 


118 58 






s = 


59 29 


1. 


cos 9.7057 


«— P=. — 


29 41 


1. 


cos 9.9389 
19.7089 


iZ = 


44 20 


1. 


cos 9.8545 


true Z = N. 


88 40 TV. 






mag. N. 


100 30 TV. 






var. 


11 50 TV. 







HOUR-ANGLE AND LOCAL TIME. 

130. Problem 42. To find the hour-angle of a heavenly 
body in the horizon. 

Solution. In the diagram of the last problem, 

MPZ = £, the hour-angle ; 
and in the triangle P M 1ST are given 
PKT^i, 
PM = 90° 



* 1 



to find MPN='180°— t. 



128 



NAVIGATION". 



Wc have 



cos M P N = 



tan P N 



tan P M' 
whence cos t = — tan d tan X. (l^ 3 ) 

131. From this it is apparent that when the latitude and 
declination have the same name, £>6 h , and consequently 
that 2 tj or the time that the body is above the true horizon, 
> 12 h ; and when the latitude and decimation are of different 
names, t < 6 h and 2 t < 12\ 

2 t is an interval of sidereal time for a fixed star, of ap- 
parent time for the sun. 

In the case of the sun, t would be the apparent time of 
sunset, were the refraction and dip nothing, and (24 h — t) 
would be the apparent time of sunrise. 

Tab. IX. (Bovvd.) contains t for different values of L and d. 

132. Problem 43. To find the hour-angle of a heavenly 
body at a given place, and thence the local time, when the 
altitude of the body and the Greenwich time are known. 

Solution. Find the declination 
of the body for the Greenwich 
time, and reduce the observed al- 
titude to the true altitude. Then 
in the triangle P Z M (Fig. 28) 
are given 

PZ = 90°-Z, 

Z M = 90° -A, 
to find 

ZPM = ^. 




For the triangle ABC (Fig. 29), we have 
j^ __ /fern (s — d) sin (a— c) 



sin-l- 



sin J) sin c 



in which, putting A — t 
a = 90° -A, 
b=p 9 

c = 90° — Z, 




HOUR-ANGLE AND LOCAL TIME. 



129 



we have s—b = 90°— |- (Z+p + h), 
s—c = \ {L+p—li), 

cos \ (L + p -f h) sin ^ (Z +p — 7i) 



and 

sin | t 
or, if we put 



/ 



cos L sin p 



s' = i (Z+p + h), 



sin -J t 



//cos «' sin (5' — Tij 



v 



(104) 



cos L sin ^? 
which is Bo wd itch's rule, p. 209. 

From Tab. XXVII. (Bowd.) we may take t directly from 
column P. M., corresponding to the log sin \ t. 

t is.— when the body is east of the meridian. 

When the object is the sun west of the meridian, t is the 
apparent solar time; when the sun east of the meridian, 
(24 h — i) is numerically the apparent time. 

"When the object is the moon, a planet, or a star, we have 
(Prob. 37), denoting its R. A. by a, 

the sidereal time = a 4- £, 
and the mean time = a—S' + t 9 

in which S f is the "right ascension of the mean sun." 
(Art. 93.) Or the sidereal time may be converted into 
mean time by one of the other methods of Problem 32. 

133. By the formula 

- . //sins sin (s — a)\ 

cos i A = A/[ — ^-tA , 

* V \ sin sin c y 

we may obtain for the triangle P Z M (z being the zenith 
distance), 

- M / [sm% (co L+p + z) sin b (co L+p — z)\ 

COS t t = A/ [ ^—. 

z V \ cos L sin p 

or putting 

s = i (co Z+p + z), 

/sin s sin (s — s)\ f (1 05) 



COS \ t = JU 



cos L sin p 
which is the rule in Bowditch's 2d Method, p. 210. 



180 NAVIGATION. 

(105) is preferable to (104) when t considerably exceeds 
6 h , which may be the case in high latitudes. 

If L =90°, the horizon and equator coincide, and 
p + h — 90° and p — z; so that both (104) and (105) be- 
come indeterminate. In very high latitudes, then, these 
equations approach the indeterminate form, and it is imprac- 
ticable to find with precision the local time from an observed 
altitude. 

So also if d = 90°, the star is at the pole and L = h; 
and the problem is indeterminate. A great declination is 
therefore unfavorable. 

134. If the object is in the visible horizon (rising or set- 
ting), h = — (33' + dip) nearly. With the sun, the instants 
when its upper and lower limbs are in the horizon may be 
noted, and the mean of the two times taken as the time of 
rising or setting of its centre. The irregularities of refrac- 
tion would affect nearly alike the dip and the apparent posi- 
tion of the sun. 

135. If the time at which the altitude is observed is noted 
by a watch, clock, or chronometer, we may readily find how 
much the watch or chronometer is fast or slow of the local 
time. (Prob. 50.) For, let 

C be the time noted, 

T, the local time deduced from the observation : 
c = T— G will be tjie correction of the watch or chronom- 
eter to reduce it to apparent time, when T is the local 
apparent time ; to mean time, when T is the local mean 
time ; or to sidereal time, when T is the local sidereal 
time. 

136. The observed altitude is affected by errors of ob- 
servation, errors of the instrument, and errors arising from 
the circumstances in which the observation is made ; such 
as irregularities of refraction affecting both the position of 
the body and the dip of the horizon. Errors of the first 



HOUR-ANGLE AND LOCAL TIME. 



131 



class are diminished by taking a number of observations. 
Thus several altitudes may be observed, and the time of each 
noted ; and the mean of the altitudes taken as correspond- 
ing to the mean of the times, so far as the rate at which the 
body is rising or falling can be regarded as uniform during 
the period of observation. This period should then be 
brief 

137. We may easily find how much a supposed error of 
V in the altitude will affect the resulting hour-angle, by di- 
viding the difference of two of the noted times by the dif- 
ference in minutes of the two corresponding altitudes. 

The effect will evidently be least when the body is rising 
or falling most rapidly. This will be the case wmen its 
diurnal circle makes the smallest angle with the vertical cir- 
cle. An inspection of the diagram (Fig. 30) shows that 
this is the case when the object is nearest the prime vertical, 
or bears most nearly east or west. 

Thus Z n being tangent 
to the diurnal circle n n\ 
the angle which it makes 
with it is ; and is there- 
fore less than the angle 
which any other vertical 
circle, as Z n\ makes with 
n 7i\ 

The diurnal circle m mil 
makes a smaller angle with 
Z m, the prime vertical, 
than with any other verti- 
cal circle, as Z m\ 
The diurnal circle o o' makes a smaller angle with Z o 
than with Z o'. 

The diurnal circles make right angles with the meridian ; 
so that at the instant of transit, the change of altitude 
is 0. 




132 NAVIGATION. 

138. At sea, and to a less extent on the land, the latitude 
is uncertain. To ascertain the effect of an error of 1' in the 
assumed latitude, the hour-angles may be found for two lati- 
tudes separately, differing, say, 10' ; and the difference of 
these hour-angles divided by 10. 

This is an essential feature of Sumner's method, which 
will be explained hereafter. This method will also show 
that an error in latitude least affects the deduced hour-angle 
when the body is nearest the prime vertical. 

Examples. (Prob. 43.) 

1. At sea, 1865, March 20, 10 h 15 m 20 s G. mean time, in 
lat. 41° 15' S., long. 86° 45' W. (by account) ; observed 
P. M. altitude of the sun's lower limb 18° 20'; index, cor. 
of sextant —8' 20"; height of eye 18 feet; required the 
local mean time. (104.) 



G. m. t. 


h m s 

Mar. 20 10 15 20 G 


's dec. Ecfn of t. 




o 


1 li li m s a 




10.256 —0 


2 3.7 + 59.23 +7 33.82 —0.754 
( 592,3 ( 7.54 




+ 


10 7.4 \ 11.8 + 7.73 \ .15 
(3.3 (4 




+ o_ 


8 4 +7 26.1 




O 18° 20' 0" 


( S. diam + 16' 5 In. cor.— 8' 20" 




+ 52 


\ par. + 8" dip. — 4 11 




7i= 18 20 52 


( ref. —2 50 




£=41-15 


1. sec 0.12387 




jp— 90 8 4 


1. cosec 




2 s = 149 43 56 






S= 74 51 58 


1. cos 9.41677 




S—h= 56 31 6 


1. sin 9.92120 




/ 


19.46184 


L. ap. t. 


Mar. 20 4 h 20 ra 28 3 


1. sin i 9.73092 


Eq. of t 


+ 7 26 




L. m. t. 


Mar. 20 4 27 54 




Subtracting the local mean time from the G. mean time 


gives 


the long. + 5 h 47 m 


2G S == 80° 51f W. If we take 



HOUR-ANGLE AND LOCAL TIME. 



133 



Z = 41° 25' S., we shall find the local ap. time 4 h 20 m 12 s ; 
so that for AZ= 10' S, A t = -16 s . 

2. 1865, Jan. 1, 21 h at the Navy-Yard, Havana, in lat. 
23° 8' 39" N"., long. 5 h 29 m 27 s W., the following altitudes 
of the sun were observed with an artificial horizon; re- 
quired the local mean time. 





T. by Chro, 


2 © 








h m s 

3 33 57.5 


5310 


Chro. fast of G. 


m s 
m. t. 42 37.7 




34 29.3 


20 


Bar. 30.43 




35 2.3 


30 


Ther. 


75° 




35 33.3 

36 47 
36 37.0 


40 
50 
60 


^i ** nm f offthearc + 32 / lS\3 

© sdiam -jonthearc-32 35.0 

Index cor. — 8. 3 


T. by Chro. 


3 35 17.35 


58 35 


©'« dec. 




Chro. cor. 


—42 37.7 


h' = 29 17 30 i 


£ In. cor. — 4" — 22 


53'42.2" + 14 ff .04 


G. m. t. Jan. 


2 3 52 39.6 = 


! 
3h.S78 + 14 42-J 


S. diam. + 16'18 
ref. — 1 40 


[42.12 

+ 54.4 ^l.| 

52 48 [ f x 






I 


par. +8—22 






7i= 29 3212' 




Eq.oft. 






Z= 23 8 29 


1. sec 0.0364304 


+4 27.S1 +1.165 






p= 11252 43 


1. cosec 0.4102711 


f 3 '^ 






2 s = 165 33 29 




+4.51 J-» 






s = 82 46 45 


1. cos 9.0993144 


+ 4 32.32 I 1 






s — h= 5314 33 


1. sin 9.9037276 
19.4497435 








it=— 32 3 16.7 


1. sin 9.724S718 




L. ap. t. Jan. 


h m 8 

1 19 41 53.7 


*=— 64 6 33.4 


= +295° 53' 26". 6 




Eq. of t. 


+ 4 32.3 








L. m. t. Jan 


1 19 46 26.0 









We have also by subtracting the chro. time from the local 
mean time, 

Chro. cor. (L. m. t.) — 6 h 12 m 4 S .8 
Long. - +5 29 27 .0 

Chro. cor. (G. m. t.) —0 42 37 .8 

As the Chro. is fast, the correction is subtractive. 



134 NAVIGATION - . 

By comparing the first and last altitudes and the corre- 
sponding times, we find that for 

2 Ah = + 50', A t = + 2 m 39 9 .5 ; or, for 2 A h— + 1', A t=z +f3M9; 

that is, an error of 1' in the double altitude will produce an 
error of 3 s in the resulting time. 

3. At sea, 1865, Sept. 7, 8 h 4 m 16 s , G. mean time, in lat. 
46° 16' N., long. 153° 0' E., the observed altitude of the 
moon's upper limb, W. of the meridian, was 21° 19'; index 
cor. of octant, —3'; height of eye 20 feet; required the 
local meau time. 

h m o / 

G. m. t. Sept. 7 8 4 16 "J 21 19 D's dec. 

In. cor. — 3 +6° 59' 34" + 1T\2 

dip. — 4 +48 44.8 

S. diam. — 17 + 7 22 2 .8 

K — 20 55 S. diam. 16' 36" + 4" 

par.&ref. + 54 H. par. 60 49 

h = 21 49 

L = 46 16 1. sec 0.16033 

p — 83 1. cosec 0.00325 

b m s s 2 5 = 151 5 

D's R. A. 11 12.2 + 2.40 s = 75 32£ 1. cos 9.39738 

+ 10.2 ~9j6 s—p == 53 43£ 1. sin 9.90644 

1 1 22 .6 19.46740 

D'sH. A. 4 22 21 1. sin j- 9.73370 

L. sid. t. 5 23 43 

_£ o —ii 6 18 

-Red.for G.m.t. — 1 20 
L. m. t. Sept. 7 18 16 5 
Long. — 10 11 49 = 152° 57' E. 



4. 1865, Sept. 30, in lat 30° 27' N"., the Chro. time of the 
setting of the sun's centre was ll h 16 m 6 s ; the Chro. cor., 
-f!5 m 25 s ; height of eye 16 feet; required the local time. 



HOUR-ANGLE AND LOCAL TIME. 135 



h m a 

T. by Chro. 11 16 6 
Chro. cor. + 15 25 




O's. dec. 

O 1 II 

2 54 31.9 


Eq'n of t. 
— 58.30 —10 4.40- 


-0.801 


G. m. t. Sept. 30 11 31 31 




— 11 11.9s 


r 641.3 ( 
29.1 -9.23-< 


8.81 
.40 


11.525 


A = - 


3 5 44 ( 
- 37 


k 1.5 -10 13.6 


2 




Lz= 


30 27 


1. sec 0.06446 






P = 


93 6 


1. cosec 0.00064 






2s = 


122 56 








s = 


61 28 


1. cos 9.67913 




s 


— h = 


62 5 


1. sin 9.94627 




h m s 

L. ap. t. Sept. 30 5 55 34 






19.69050 
1. sin \ 9.84525 




Eq. of t. — 10 14 










L. m. t. Sept. 30 5 45 20 


Long 


+ 5 h 30 ra 46 s = 82° 4l'i W. 





139. Problem 44. To find the hour-angle of a heavenly 
body when nearest to, or on, the prime vertical of a given 
'place. 

Solution. If d > X, and with the same name, as for the 
body whose diurnal path is n n' (Fig. 30), P Z n will be 
greatest, or nearest to 90°, when Z n is tangent to n n\ and 
consequently Znp = 90°. We then have 

cos t = -—4 = i r ( 106 ) 

cot L tan a v J 

If £?<X, and with the same name, as. for the body whose 
diurnal path is m m', the body will be on the prime vertical 
at m, and PZm=90°; whence we have 

tan d . „ . x 

If d and X are of different names, the diurnal circle inter- 
sects the prime vertical below the horizon, and the visible 
point nearest the prime vertical is in the horizon. The 
hour-angle of this point can be found by (104), omitting the 
effect of refraction, 



136 NAVIGATION. 

cos t = — tan d tan JO. 

Altitudes less than 8°, however, are to be avoided. 

If d = Jj, the diurnal circle passes through the zenith, and 
the body would be on the meridian and prime vertical at 
the same instant ; so that, when d and L are nearly equal, 
altitudes observed within a few minutes of the meridian 
passage of the body may be used for finding the time. It is 
only necessary that the change of altitude shall be sufficient- 
ly rapid. 

But when the body is very near the meridian in azimuth 
the change of altitude is proportional, not to the intervals 
of time, but to the squares of the hour-angles. (Art. 150.) 
Hence, when the body is in such a position, the mean of 
several times does not correspond to the mean of the alti- 
tudes. 

From the hour-angle the local time may be found by Pro- 
blems 36 and 37. 



Examples. (Prob. 44.) 

1. Find the time of the greatest eastern elongation of 
Polaris, 1865, July 16, in latitudes 50° 18' N. and 10° 27' N. 
and long. 58° W. 

L = + 50° 18' 1. tan 0.0808 L = + 10° 27' 1. tan 9.2659 

d= + 88 35.2 1. cot 8.3922 d=+88 35.2 1. cot 8.3922 

t = — 5 h 53 m .2 1. cos 8 .4730 t = — 5 h 59 m .O 1. cos 7.6581 
* R. A. = 1 10 .3 1 10 .3 

L. sid. t. 19 17 .1 19 11 .3 

— aS^ <— 7 39 .9 — 7 39 .9 

L. ra. t. 11 37 .2 11 31 .4 



2. Find the time when the sun is on the prime vertical, 
1865, June 10 A.M., in latitude 26° 15' N., longitude 
155° 16'E.= — 10 h 21 m 4 9 . 



HOUR- ANGLE AND LOCAL TIME. 137 

X=4-26°15' 1. cot 0.3070 Q , sdec. + 23° 2' 33" +11'.3 

rf= + 23 0.6 1. tan 9.6281 -157 113 

4 

*=— 2 h 2 m .l 1. cos 9.9351 23 36 

L. ap.t. June9 21 57 .9 Eq'noft. — m 53 s + (K49 

Eq'n of t. —0 .9 —5 

L. m. t. June 9 21 57 .0 or June 10 9 h 57 m .O A.M. — 55 

3. 1865, June 25, in lat. 40° 15' N., long. 65° 17' W., re- 
quired the times when a Lyras and a Aquilae are on the 
prime vertical. 

a Lyrce. a Aquilce. 

L = + 40° 15' 1. cot 0.0723 L = + 40° 15' 1. cot 0.0723 

d— +38 40.8 1. tan 9.9034 d— + 8 31.1 1. tan 9.1754 

t— T l h 16 m .O 1. cos 9.9757 t = T 5 h 19 m .2 1. cos 9.2477 

a= 18 32 .4 a— 19 44 .2 

L. sid. t. 17 16 .4 or 19 h 48 m .4 14 25 .0 or l h 3 m .4 

-S'° -6 17 .1 -6 17 .5 -6 16 .6 -6 18 .4 

L.m.t. June 25 10 59 .3 13 30 .9 June 25 8 8 .4 18 45 .0 



CHAPTER VII, 



LATITUDE. 



140. Problem 45. To find the latitude from an observed 
altitude of a heavenly body on the meridian. 

Solution. Let the diagram (Fig. 31) be a projection of the 
sphere on the plane of the meridian NZS: 

Z, the zenith ; 
N S, the horizon ; 
P, the elevated pole ; 
P P', the axis of the sphere ; 
E Q, the equator ; 
Q Z, the declination of the ze- 
nith, and 
NP, the altitude of the pole, 
are each equal to the lati- 
tude, L. 




Fig. 81. 



Let 



M be the position of the body ; 

Q M = c?, its declination ; 

MZ = 2 = 90°— A, its zenith distance, which it is conven- 

k at to mark N". or S., according as the zenith is north or 

south of the body. 

From the diagram, we have Q Z = Q M + M Z 
or, Z = z + d, (108) 

which is the general formula. 



LATITUDE. 139 

If the body is at M', numerically 
Z* = z — d; 

ifatM", £=d-z; 

or " the latitude is equal to the sum of the zenith distance 
and declination, when they are of the same name ; to their 
difference, when of different names ; and is of the same name 
as the greater." (Bowd., p. 166.) 

If the body is at M'", or below the pole, 

QM r// =180 o -c?, and L = 180°- d- z, 
numerically ; or (108) is the correct formula, provided we 
use 180°— c?, or the supplement of the declination, instead 
of the declination. 

But in this case we have also from the diagram 

Z=p + h, (109) 

as in Bowditch, p. 167. 

The declination of the body must be found from the Al- 
manac for the time of meridian passage. (Probs. 23, 27.) 
The observed altitude must be corrected for dip, refraction, 
&c. r and the true altitude derived, 

From (108) we see that an error of 1' in the altitude will 
produce an error of 1' in the resulting latitude. 

Examples. 

1. At sea, 1865, June 30, in lat. 2^° N., long. 105° 18' W., 

the observed meridian altitude of the sun's lower limb was 

69° 15' 20", sun bearing N". ; index cor. +3' 20" ; height of 

eye 20 feet ; required the latitude. * 

Long. + 7 h l m 12 s r=7 h .02 

_0 69° 15' 20" fin. cor. + 3' 20" 

J S. diam. +15 46 O's dec. 23° 10' 21" N. — 9".29 
+ 14 24 I Dip - 4 24 _ x 5 ^~ 

Uef.&p.- 18 d=n 916N 

k - 69 29 44 z= 20 30 16 S. 

£= 2 39 ON. 



140 NAVIGATION. 

2. At sea, 1865, June 30, in lat. 43^° N., long. 150° 15' E 
0= 69° 15' 20" ; on meridian bearing S.; index cor. +3' 20"; 
height of eye 20 feet ; required the latitude. 

Long. — 10 h l m s = — 10 h .02 

69° 15' 20" fin. cor. + 3' 20" 

J S. diam. +15 46 O's dec. 23° 10' 21" N. -9".29 
+ 14 24 j Dip — 4 24 +1 33 

[Ref. &p.- 18 d=23 11 55 N. 

h = 69 29 44 z = 20 30 16 N. 

Z = 43 42 11 N. 

3. At sea, 1865, Aug. 13, 5 A.M., in lat. 25° S., long. 
85° 15' W., obs'd mer. alt. of D's U. limb, 50° 18'; moon 
north ; index cor. —2' ; height of eye 16 feet ; required the 
latitude. 

D's S. diam. 16' 7" +12" 
D's H. par. 59 4 



Long. 




+ 5 h 41 m s =r5 h .68 


J)'s mer 


pass 


Aug. 12 17 11.4 +2 m .28 


Red. for 
L. m. t. 


long. 


( 11.40 
+ 13.0 1 1.37 

( 18 
Aug. 12 17 24.4 


G. m. t. 




Aug. 12 23 5.4 


D's dec. 




14° 29' 35"N. + 7".44 
d=zU 30.3 N. 










z = 39 27.1 S. 






L = 24 56.8 S. 



3" 50° 18' (In. cor. — 2'.0 

— 22 3"| S - diam. — 16.3 

(Dip - 4.0 

7t'=49 55.7 

+ 37.2 Par. and ref. 
£=50 32.9 



4. At sea, 1865, Oct. 9, 5 P.M., in lat. 65|° N., long. 
150° E. ; obs'd mer. alt. of a Lyrie 63° 17', bearing S. ; index 
cor. +3' 30" ; height of eye 17 feet ; required the latitude. 

*'salt. 63° 17' (In. cor. + 3'.5 
—1 } Dip -4 .0 
h = 63 16 ( Ref - - - 5 
z = 26 44 N. 

d = 38 40 N. 
L = 65 24 N. 

If the star bore N., the latitude would be 11° 5Q f 1ST. 



LATITUDE. 141 

5. At sea, 1865, June 18, in lat. 23j° N"., long. 163° 0' E. ; 
obs'd mer. alt. of Q's 1ST. limb from X. point of the horizon, 
89° 50'; index cor. -f 1' 20"; height of eye 21 feet; required 
the latitude. 



Long. — 10 h 52 m 9 = — 10 h .87 


0's dec. 23° 25' 27" N. + 3".0 


O 89° 50' ( In cor. + l'.S 


— 33 


\ S.diam.+15.8 




+ 12,6 (Dip — 4.5 


d = 23 24.9 N. 


h = 90 2.6 


z= 2.6 N. 




L — 23 27.5 N. 



In this example, the true altitude of the O's centre is 
more than 90° ; this changes the sign of z. 

6. At sea, 1865, May 18, in long. 180° 0' E., the true mer. 
alt. of the sun was 75° 18'; sun bearing S. ; required the 

latitude. 

Long. — 12 h 0' " d = 19° 30' N. 

z = 14 42 N. 
Z= 34 12 K 

7. At sea, 1865, May 17, in long. 180° 0' W. ; the true 
mer. alt. of the sun was 75° 18 r ; sun bearing S. ; required 
the latitude. 

Long. +12 h m 9 d = 19° 30' N. 

z = 14 42 N. 
L = 34 12 N. 

Examples 6 and 7 are identical, the Greenwich apparent 
time being May 17 12 h for both. They illustrate the. neces- 
sity as well as propriety of the rule for navigators near 
the meridian of 180°, to add l d to the date, Avhen they pass 
from west longitude to east ; to subtract l d from the date, 
when they pass from east longitude to west. For instance, 
May 18 5 h in long. 180° 15' E., is identical with May 17 
5 h in long. 179° 45' W. 

141. The common mode at sea of measuring a meridian 
altitude of the sun, is to commence observing the altitude 
20 or 30 minutes before noon, repeating the operation until 



142 NAVIGATION. 

the highest altitude is attained ; soon after which the sun, 
as seen through the sight-tube of the instrument, begins to 
dip, or descend below the line of the horizon. 

It is preferable, however, to find, from A.M. observations 
for time and by allowing for the run .of the ship in the inter- 
val, the time of apparent noon by a watch, and observing the 
altitude at that time within l m or 2 m . 

A meridian altitude of the moon, or a star, can be much 
more conveniently observed by finding beforehand the watch 
time of its culmination, and measuring the altitude at or 
very near that time. 

When the sea is heavy, it is recommended to observe 
three or four altitudes in quick succession, within 2 m of the 
time of culmination. 

142. If the body is changing its declination, or the ob- 
server his latitude, the maximum altitude is not at the in- 
stant of meridian passage ; but after, if the body and zenith 
are approaching ; before, if they are separating. Let 

t be the hour-angle of this culminating point, in minutes / 
A d, the combined change* of declination and latitude in l m , 

if it is expressed in seconds; or in l h , if it is expressed in 

minutes / 
A h, the change of altitude in l m from the meridian passage 

due solely to the diurnal rotation, (from Tab. XXXII.) ; 
Ah, the reduction of the maximum altitude; both expressed 

in seconds. 

• Now in the time t 

t A (7 will be the excess of altitude produced by the change 
of declination and latitude ; 

f A h (as will be shown in Art. 150), the diminution of alti- 
tude due to diurnal rotation ; 

* Their sum, if they both tend to elevate or both to depress ; otherwise 

their difference. 



LATITUDE. 143 

and we shall have 

Ah = tAd-FA h. 

But at a point whose hour-angle is 2 t, the altitude will be 

the same as the meridian altitude, or 

= 2tAd — (2t) 2 A h; 

whence t — zrTTi 

2 A h 

and •Ak = itAd*ffi j 

which accord with the rule in Bowditch, p. 169. 

Example. A ship in lat. 62° 1ST., on March 21, sails south 
14 miles per hour. 

Ad — 14' + 1'= 15' per hour, or 15" per* minute 



(110) 



4>A = 


1\0 ; 




t- 


15 m _ 


7* m 


Ah = 


is 
T x] 


5"= 



56". 

The uncertainty of altitudes at sea makes such a correc- 
tion of little practical importance; but it is generally ne- 
glected by those navigators who work out their latitudes to 
seconds, supposing that they have attained that degree of 
accuracy. In the above example, the maximum altitude of 
the sun would have been greater than the meridian altitude, 
and the latitude obtained from it in error, by nearly 1'. The 
sun would not have sensibly dipped until 9 or 10 minutes 
after noon. 

143. A difficulty occurs at sea in measuring the meridian 
altitude of the sun when it passes near the zenith, on account 
of its very rapid change of azimuth ; the change being made 
from east to west, 180°, in a very few minutes. 

What is wanted is the angular distance of the sun from 
the'N". or S. points of the horizon. One of these points may 
be sufficiently fixed by means of the compass, and then the 
angular distance from this point observed within l m or 2 m of 



144 



NAVIGATION.. 



the meridian passage as determined by a watch regulated 
to apparent time. 

144. From (108) we have 

z — Z — d, (111) 

by which the zenith distance maybe found when the latitude 
and declination are given. 

Also d = Z — 2, which may be used at sea for estimating 
the declination of a bright star from its estimated meridian 
altitude. If the time when it is near the meridian be also 
noted, and converted into sidereal time, we have the right 
ascension and declination of the star sufficiently near for de- 
termining what star it is. 

Example. 

July 16, 8 h 45 m , in lat. 11° N"., a bright star is seen near 

the meridian S., at an estimated altitude of 55°. 

L. m. t. July 16 8 h 45 ln L = 11° N. 

S Q V 37 2 = 35 N. 

L. sid. t. 16 22 d=24 S. 

The R. A. of a Scorpii {Antares) is 16 h 21 m , and its declina- 
tion 26° r S. 

145. Problem 46. To find the latitude from an altitude 
of a heavenly body observed at any time, the local time of 
the observation and the longitude of the place being given. 

1st Solution. Reduce the observed altitude to the true, 
altitude, and from the local time 
and longitude find the declina- 
tion and hour-angle of the body. 
(Probs. 21, 34, 35.) Then in the 
triangle P Z M (Fig. 32) there 
are given 

ZPM = £, 

PM = 90°-e?, 
ZM = 90°— A, 
to find 

PZ = 90°-Z. 




LATITUDE. 



145 



By Sph. Trig. (146), if in the triangle ABC (Fig. 33) are 

given a, 5, and A, we find c by the formulas 

tan (ft = tan b cos A, ^ 

. . cos cos a 

cos (ft = z — t — ■ 

r COS 

c = (ft ± 0' ; 

which, applied to the triangle 
P Z M, give 




Fig. 33. 



tan (ft = 

COS <£'- 



cot c? COS £, " 
cos-0 sin A 



(112) 



sin d 
90°— Z = 0=b0'. 

These may be changed into a more convenient form for 
practice, if we put </> = 90°— 0" ; then 
tan r/ = tan d sec £, 
sin 0* sin h 



COS 0'= 



(113) 



sin a 

Here, observing that -f- and -— may be rendered by X. 
and S. respectively, we mark (ft" N". or S. like the declina- 
tion, and (ft' either X. or S. ; then the sum of (ft" and (ft' when 
of the same name, their difference when of different names, 
is the latitude, of the same name as the greater. There are 
two values of L corresponding to the same altitude and 
hour-angle, but which, imless (ft' is very small, will differ 
largely from each other ; so that we may take that value 
which agrees best with the supposed latitude (at sea the 
latitude by account). When t > 6 h , (ft" > 90°, as in (97). 

146. In Fig. 32, if Mm be drawn perpendicular to the 
meridian, we shall have 

eft = P m, the polar distance of m, 

<ft fr = 90°— P m, the declination " 

<f>'=Z m, the zenith distance " 



146 NAVIGATION. 

When </>' is very small, (that is, when M m nearly coin- 
cides with M Z), (/>' cannot be found with precision from its 
cosine. If not greater than 12°, it can be found only to the 
nearest minute with 5-place tables ; if only 2°, it can be 
found only within 3'. The more nearly, then, that M m co- 
incides with Z m, or, in other words, the nearer the body is 
to the prime vertical, the less accurate is the determination 
of the latitude. If the body is on the prime vertical, cos </>'=. 1, 
and <// cannot be found within 30'. 

147. To find the effect of an error in the altitude, let 

A h = a small change of altitude ; 

A cj) r = the corresponding change of <j> f ; it will also be numer- 
ically the change of latitude, as (f> n does not depend on h; 
Then from the 2d of (113) 

or, since A h and generally A <p> are so small that we may 

take 

cos A h—l, sin A h =A h sin 1", 

cos A 0'= l, sin A <j> r =A 0' sin 1", 

coscb'—Ad)'. sin 6' sin V~ . ., (sin h + A h. cos h sin 1"). 

T T T sin a v ' 

Subtracting this from the second of (113), and reducing, we 
have 



or, since 



. ,, sin <p n cos h . , 

sin a sin f * 



cos 6 r sin (j)" 



sin h sin d ' 
A <£'= _ A h. cot <f>' cot A. (114) 

But in the triangle M Z m, 

lrr7 t» r r^ t^ tan m Z 

cos M Z ra = — cos M Z P = z — tf~ ; 

tan M Z 7 

that is, Z being the azimuth, 



LATITUDE. 147 

— cos Z = — — t-, or sec Z = — cot 0' cot A, 
cot A 

and therefore 

J </>'= J A. sec Z. (115) 

If the body is on the meridian, Z— or 180°, and numer- 
ically A (p f =A h. 

The nearer Z is to 90°, the greater is A 0'. If Z = 90°, . 
or the body is on the prime vertical, sec Z = oo , and J 0' is 
incalculable. If Zis near 90°, (115) is inaccurate; since A <J> r 
becomes too large for the assumptions 

cos A 0'=1, sin A <f>'=A (/>'. sin 1" ; 

so, also, in (114) if <p! is very small, A $ may become large. 
A star which transits the meridian near the zenith, changes 
its azimuth very rapidly. Unless observed on the meridian, 
it cannot be depended on for latitude. 

148. To find the effect of an error in the time, and con- 
sequently in the hour-angle, we may take the formula in 
Prob. 39. 

sin h = sin L sin d 4- cos L cos d cos t, (H6) 

and letting 
A t = a small increase of the hour-angle, expressed in time, 

and 
JZ = the corresponding change of the latitude, by a similar 
but more complicated process,* we shall obtain 

JJ= _15co a Jco S »'tant 

sin f v ' 

But, from (97), changing the notation and regarding Z as 

positive toward the right, 

„ cos <£" tan t " 
tan Z = - 

and by substitution in (116), 



sin p' } 



i 



* Such formulas can be more simply obtained by the process of differen- 
tiation. 



148 NAVIGATION. 

A L = — 15 A t. cos L tan Z, (118) 

which requires that the azimuth should be known. 

At sea the chief uncertainty of this problem is in the time, 
either from its imperfect determination by observation, or 
from unavoidable errors in allowing for the run of the ship 
in the interval between the observations for time and for 
latitude. 

By (118) it appears that the effect of an error in the time 
is when Z = or 180°, that is, when the body is on the 
meridian ; and the effect is incalculable, when Z n 90° or 
270°, or the body is on the prime vertical. 

Moreover the effect is opposite on different sides of the 
meridian, and would be eliminated by two observations of 
the same body, or of different bodies, at the same azimuth 
E. and W. of the meridian. 

149. 2c? Solution. If the latitude is already approximately 
known, we have, (116) 

sin h ±= sin L sin d + cos L cos d cos t ; 
whence 

cos (i — d = sin h + 2 cos L cos d sin 2 \ t; 

or since {L — d) is the meridian zenith distance of the body 
(108), denoting it by £ , and the meridian altitude by A , we 
have 

cos z Q — sin h = sin h + 2 cos L cos d sin 2 \ tA 
or V (118) 

cos z = sin h — sin h 4- cos L cos d versin t / ) 

in which we may use the approximate value of L in com- 
puting the term cos L cos d versin t ; which term is smaller 
the nearer the observation is taken to the meridian. Having 
found the meridian zenith distance, we may find the lati- 
tudes as in Prob. 45. If the computed value of L differs 
largely from the assumed value, the computation should be 
repeated, using this new value. 

This is the method in Bowditch, page 200. 



LATITUDE. 149 

150. 3d Solution. When the observation is taken very 
near the meridian, we may find the correction to be applied 
to the observed altitude to reduce it to the meridian altitude, 
thus : 

From (118) we have 

sin A — sin A = 2 cos L cos d sin 2 \ t, 
whence, by Sph. Trig. (106), 

cos \ (A 4-A) sin -J- (k —h) = cos L cos d sin 2 J t. 
But A and /*, differing very little, we may put 

cos | (A +-A) = cos A =sin £ == sin (L—d), 

so that 

, /T TN cos Z cos d sin 2 -J- (5 /. , ^x 

sin | (A.-A) =————-. (119) 

Put A h = A — A, the reduction of the observed to the 
meridian altitude, or, as it is usually called, " the reduction 
to the meridian ;" and, since A h and t are quite small, put 

gin \ A h = ^ A 7i. sin 1", (A A being expressed in seconds of arc), 
sin | $ = £*Xlosinl", (t " " " of time), 

then (119) reduces to 

, , 112.5 sin V cos L cos d .„ 

Ah — r-jf — -^ X t* : 

sm (L — a) * 

or, since sin 1" == 0.000004848, 

' 7 0". 000545 cos L cos d „ , . ... 

A h = -r-. — f 37 X v (t in seconds). 

(sin L — a) v J 

In this formula t is in seconds of time ; but if, as is usual, 

t is expressed in minutes, we must put (60 i) 2 for f, so that 

we have 

i 7 1 ".96349 cos L cos <Z . -' 

^ A = r— T ^ X * 2 (120) 

sm (L — d) v y 

If t = l ra , the formula expresses the change of altitude in 
one minute from the meridian. Representing this by A A, 
we have 



150 NAVIGATION. 

, 7 1".96349 cos L cos d , . 

4, h — r— 7j 7 t (121) 

sm (L — d) v ' 

J h = t\AJ h \ 

and [• (122) 

/? = A-f- A h, the meridian altitude. ) 

"Whence the latitude is found as by a meridian altitude 
(Prob. 45). 

Bowditch's Tab. XXXII. contains the values of A h for 
each 1° of declination from to 24°, and each 1° of latitude 
from to 70° ; except when L—d < 4°, for then A h is so 
large that (120) and (121) become inaccurate. In this case 
the body is near the zenith, and altitudes out of the meri- 
dian do not afford a reliable determination of the latitude. 

Bowditch's Table XXXIII. contains f for each l 9 of t 
from to.l3 m . 

When h is small, the reduction to the meridian may be 
found by this method quite accurately even when t is as 
great as 12 m . If h is near 90°, t must be taken within much 
narrower limits. (Bovvd., p. 202.) Indeed, in this case z , 
or its equal (i— c7), is very small, and consequently A h 
becomes large. Unless then t is sufficiently small, A h will 
be too great for the assumption sin f A h = A h sin V. 

If d > X, sin {L—d) == sin z is negative ; that is, z will 
have a different name or sign from L (Art. 140). Properly 
A, 7i , and A h would also become negative to correspond. 
Still, however, we shall have numerically 

7i = h + A h. 

We may therefore disregard the sign of L—d in (121) 
and consider h and h as always positive. 

If the star is observed at its lower culmination, then t 
will be the hour-angle from the lower branch of the meri- 
dian, and for d we may use 180°— d (Art. 140). A Q h and A h 
are then numerically subtractive. 



LATITUDE. 



151 



Examples. (Prob. 46.) 

1. At sea, 1865, July 17 l h P. M., in lat. 36° 38' S., long. 
105° 18' E., by account; time by Ckro., 5 h 47 m 14 s ; O., 
30° 15'; N". W'y; index cor. + 2' 30"; height of eye, 17 feet ; 
Chro. cor. (G. m. t.) + I4 m 3 s ; required the latitude. 







By (113) 


h m s 

T. by Chro. + 12k, 17 47 14 
Chro. cor. +14 3 


G*s dec. Eq'n of t 

G / // // m 3 s 

+ 21 20 29 —25.19 —5 43.8 —0.230 


G. m. t. 


July 16 18 1 11 = 


18.021 -7 34 ( 252 -4.1 j 2.3 
+ 21 12 55)202 _ 3 47.9 { 1-8 


—Long. 


+ 7 1 12 

July 17 1 2 29 

— 5 48 


L. m. t. 

Eq. of t. 


_0 30° 15' ( In. cor. + 2'. 5 dip. — 4'.0 
+ 13 ( S.diam + 15 .8 ref.&par. — 1 .S 


L. ap. t. 


56 41 


h = 30 28 1. sin 9.70504 




o / // 

** = 14 10 15 


1. sec 0.01342 




d = 21 12.9 X. 


1. tan 9.58903 1. cosec 9.44145 




f = 21 49.1 N. 


1. tan 9.60245 1. sin 9.57016 




$' - 58 37.0 S. 
L - 36 48 S. 


1. cos 9.71665 







If we suppose an uncertainty of 3' in the altitude and 20' 
in the longitude, by (115) and (118) 



Z=S. 164° 40' W. 
A h = + 3' 
AL--ZW 



L cot (— h) 0.2304 a 1. cos L 9.903 

Lcotf 9.7853 —A t= — 20' log 1.301 n 

1. sec Z 0.0157 n 1. tan Z 9.438 n 

log 0.477 jZ=+4'.4 1og 0.642 

log 0.493 n 



That is, an increase of 3' in the altitude will numerically 
decrease the latitude 3'.1 ; and a numerical increase of 20' 
in the assumed longitude will increase the latitude 4'. 4. This 
may be conveniently expressed in the following way : 



* Instead of changing t into arc, we may enter col. P. M. of Tab. XXVII. 
with 2 t = l h 53 m 22 9 . 



152 



NAVIGATION. 



Long. 105° 18' 



20' E.; O, 30° 15' =fc 3' 
L = 36° 48' ± 4 '.4 zp 3'.1 S. 

By (119) 



t = h 56 m 41 a 


2 1. sin £ 8.18227 ) 


^or i. versin 


d— 21° 12'.9 


N. 


1. cos 9.96953 
8.45283 




1st L — 36 38 


S. 


1. cos 9.90443 








log 8.35726 


.02276 


h = 30 28 






sin .50704 


Z = 58 .5 


S. 




cos .52980 


2d L — 36 47 .6 


s. 


1. cos 9.90345 








log 8.35628 


0.2271 


Z =:58 0.7 


s. 




cos .52975 


3d i = 36 47 .8 


s. 







2. At sea, 1865, Jan. 5, 6 h P. M., in lat. 50° 36' K, long. 
135° 25' W. (by account), time by Chro. 3 h 10 m 15 s ; Chro. 
cor. (G. m. t,)— 18 m 56*; Obs'd alt. of Mars, 45° 18'; 
S. E'y; index cor.- — 3'; height of eye, 19 feet; required 
the latitude. (113) 



T. by Chro. + 12\ 15 10 15 

Chro. cor. —18 56 

G. m. t. Jan. 5 14 51 19 = 14\855 

S 19 22.1 

Red. for G. m. t. 4- 2 26.5 
G. sid. t. 9 54 7.6 

—Long. —9 1 40 

L. sid. t 52 27.6 ff = 45° 18' 

Mara 1 R. A. 3 55 25.6 — 8 

/=- 3 258 h =45 10 

or 45° 44' 30" 
d = 23 37 N. 
<p" = 31 19.3 N. 
f = 19 25.5 N. 
L- 50 45 N. 



Mars' R. A. 



3 55 25.1 

+ 0.5 

3 55 25.6 



+ 0.037 



f: 



37 
14 



Mars' dec. -f 23° 0' 30" 

+ 7 

+ 23 37 



0".50 



1. 


sec 0.15621 


1. 


tan 9.62807 


1. 


tan 9.78428 



( In. cor. — 3' 
( dip. & ref. — 5 

1. sin 9.85074 



1. cosec 0.40793 
1. sin 9.71588 
1. cos 9.97455 



LATITUDE. 



153 



If Ah— + 5' and A 2,= + 15', J *= —15'; and by (115).and (118) 



1. cot (—h) 9.9915 n 
1. cot <j>' 0.4527 
Z— X. 110° 46' E. 1. sec Z 0.4502 n 
J h=+ 5' log 0.699 
Ji- - 14'.1 log 1.149 n 



1. cos L 9.801 
—J *= + 15', log 1.176 
1. tan Z 0.421 n 
A £= — 25'.0, log 1.39S n 



3. 1865, Feb. 17, near noon, at the light-house, TV. end of 
St. George's Island, Apalachicola Bay, long. 85° o lb" W. ; 
5 observations with sextant No. 1, art. hor'n No. 3, A end 
toward observer : 

T. by Chro. h 16 m 21 8 .6; 2© 98° 14' 44", (S.) ; in. cor. +2' 30"; 
Chro. cor. (L. m. t.) — 18 m 30 s A ; Bar. 30.48, Ther. 43°. 



T. by chro. h 16 m 21 s .6 

Chro. cor. —18 30 .4 

L. m.t.Feb. 16 23 57 51 .2 

Long. +5 40 21 

G. m.t.Feb. 17 5 38 12 = 

Eq. oft. —14 14 .1 

L. ap. t. ( 23 43 37 .1 

or 

—0 16 22 .9 

i— 4° 5' 43".5 
d = — 11 46 55 .7 
(j)"= — 11 48 41 .2 
(j>'= + 41 26 10 .2 
L= + 29 37 29 



By (113) 

O's dec. 
— 11° 51' 52".8 
+4 57 .1 
—11 46 55 .7 
= 5*637 
©48 



Eq.oft. 
-52".71-14' 15".22 + s .2O4 

( FT020 
+ 1.15-1 .122 
( 6 

-14 14 .07 




7' 22" f \ In. cor. + V 15" 
j S. diam. +16 30 
+ 16 57 1 Ref. — 54 

[Par. + 6 

A=48 24 19 



1. sec 0.0011104 
1. tan 9.3192842 n 
1. tan 9.3203946 n 



I. sin h 9.8738198 
1. cosec 0.6899636 n 
1. sin 9.3111004 n 
1. cos 9.8748838 



By (121) and (122) 



l s .96349 

L= + 29° 37 

d = — ll 47 

L — d— +41 24 

AJi=- 2".527 

* = — 16 m .382 
A h ' = + 678".l 



log 0.2930 

1. cos 9.9392 

1. cos 9.9908 

1. cosec 0.1796 

log 0.4026 

2 log 2.4287 

lo* 2.8313 



h= 48° 24' 19" 
A Q h= + 11 18 
h =z 48 35 37 
Z Q = + 41 24 23 
d= — ll 46 56 
L= + 29 37 27 



151 NAVIGATION. 

LATITUDE BY CIRCUM-MERIDIAN ALTITUDES. 

151. Problem 47. To find the latitude from a number of 
altitudes observed very near the meridian, the local times 
being k?ioio?i. 

Solution. By (122) we see that very near the meridian 
the altitude of a body varies very nearly as the square of its 
hour-angle. Hence we cannot regard the mean of several 
altitudes as corresponding to the mean of the times, since 
this is assuming that the altitude varies as the hour-angle, 
Let 

/*!, A 2 , A 3 , &c, be the several altitudes ; 
t u t 2 , £ 3 , &c, the corresponding hour-angles expressed in 
minutes; 

and we have as the reduction of each altitude to the merid- 
ian, and the deduced meridian altitude, 

A l h = t\. A h h — h x + ^A) 

A 2 h — tl.A Q h h = h 2 + A 2 h > &c. (123) 

4 h — tl.A h &c. h = h 3 + 4 h ) 

Thus the meridian altitude may be derived from each alti- 
tude, and the mean of all these meridian altitudes taken as 
the correct meridian altitude. But the following is a more 
expeditious method : — 

If n is the number of observations, the mean value of 
h Q will be 

7 h 1 + h i + h 9 + ...h n , AJi-\-AJi + AJi-\-...A n h 

n — _ — 

u n n 

or, 

A I + A a + ^ + . : j, g ± j + q + ...g ( 

n n v J 

Whence the rule : — 

Take the mean of the squares of the hour-angles in minutes 
(Tab. XXXIII., Bowd.) ; multiply it by the change of alti- 
tude in l m from the meridian (Tab. XXXII.) ; and add the 



LATITUDE BY CIRCUM-MERIDIAN ALTITUDES. 155 

product to the mean of the altitudes. The result is the mean 
meridian altitude required. (Bowd., p. 201.) From the 
meridian altitude thus found deduce the latitude as from 
any other meridian altitude. (Prob. 45.) Strictly, however, 
the declination to be used is that which corresponds to the 
mean of the times, and the hour-angles, £, are intervals of 
apparent time for the sud, and of sidereal time for a fixed 
star. , 

152. It is unnecessary to reduce each observed altitude 
separately to a true altitude ; as the reductions, excepting 
slight changes of refraction and parallax, are the same for 
all, and may be computed for the mean of the observed 
altitudes, and applied to this mean with the reduction to 
the meridian. 

153. Should it be desirable to compare the several obser-* 
rations with each other, and test their agreement, it will be 
sufficient to compute the several reductions to the meridian, 
A y A, J 2 hj 4* A, &c, and apply them separately to the read- 
ings of the instrument ; or to the half-readings when the 
altitudes are observed with an artificial horizon : applying, 
also, the semidiameter when both limbs of the body are 
observed. 

154. If the altitudes are taken on both sides of the merid- 
ian, and at nearly corresponding intervals, a small error in 
the local time will but slightly affect the result ; for such 
error will make the estimated hour- angles and the corre- 
sponding reductions on one side of the meridian too large, 
and on the other side too small. (Bowd., p. 203.) 

155. This method is rarely used at sea, as a single altitude 
on or near the meridian suffices. N"o increase of the number 
of observations will diminish at all the error of the dip, which 
affects alike each observation and the mean of all.* But on 

* Such an error is called constant ; those which affect the several obser* 
vations differently are called variable. 



156 NAVIGATION. 

land it is preferable to measure a number of altitudes at the 
same culmination of the body, and thus diminish the " error 
of observation." Altitudes of the sun are used, but the best 
determinations are from the altitudes of a bright star. To 
facilitate the operations, and avoid mistaking one star for 
another, it is well to compute the altitude approximately be- 
forehand. (Art. 144.) 

If an artificial horizon is employed, the error of the roof 
is partially eliminated by making two sets of observations 
with the roof in reversed positions. 

156. If two stars are observed which culminate at nearly 
the same altitude, one north, the other south of the zenith, 
the error of the instrument is nearly eliminated ; for such 
error (except accidental error of graduation) will make the 
latitude from one of the stars too great, and that from the 
'other too small by very nearly the same amount ; the more 
nearly, the less the difference of the altitudes. The error 
peculiar to the observer is also eliminated. 

If the observations are made with an artificial horizon, 
the error of the roof is eliminated, if the same end is 
toward the observer in both sets of observations. 

157. Bowditch's Tab. XXXII. extends only to d = 24° 
If a star is used whose declination is beyond this limit, or 
if greater precision than the table affords is required, A Q h 
may be computed for the star and place by (121). 

a l ;/ .9635 cos L cos d 

J ° /l " lin (L-d~)~ 

158. If the observations are made at the lower culmina- 
tion of the star, we have only to use in the formulas I80°—d 
instead of d. (Art. 140.) 

159. The altitudes observed at the same culmination are 
very nearly the same. To render the measurements inde- 
pendent, after each observation move slightly the tangent 
screw of the instrument. With the sextant, it is best to 



LATITUDE BY CIRCUM-MERIDIAN ALTITUDES. 157 

make the final motion of the tangent screw at each observa- 
tion always in the same direction, for example, in advance. 

Examples. (Prob. 47.) 

1. 1865, Jan. 10 h . Circum-meridian altitudes of Q ob- 
served at the Custom-House, Key West, Florida, 45" ]ST. of 
Light-House: lat. 24° 34' N., long. 81° 48' 31" W. 

T. by Watch. Sextant No. 1. Art. hor. No. 1. 

m 5 2 o 87 41 30 A end of Hor. h ^TTT . 

5 30 41 30 Chro. 1843 4 58 24 6 26 41 

5 50 41 35 Watch 11 39 20 1 7 30 

7 8 4140-5. C— W +5 19 4 5J_9 11 

7 30 41 50 Chro. cor. (L. m. t.) — 5 h 18^~34M) 
7 55 41 55 

9 2 O 86 36 25 _ r off arc ^ 4r>0 

9 25 36 40 U S aiam * | on arc 32 6 .5 

9 45 36 30 In. cor. + 20 .2 

12 2 35 20 A. " 

12 25 34 50 Bar - s 0.21 

12 45 3 4 40 T k ei \ 78° 

h m s o / // 

T. by W. 8 41.2 2 "o 87 41 40 

W. cor (L. ap. t.) -7 30.1 2 O. 86 35 44 
L.ap. t. Jan. 10 1 11.1 2 O 87 8 42 
Long. +5 27 14.5 

G. ap. t. Jan. 10 5 28 25.6 (At h 9 m ) C— W. . 5* 19 m 6 3 .3 

5.474 Chro. cor. (L. m. t.) —5 18 34 .0 

W. cor. (L. m. t.) + 32 .3 

Eq'n of t. +7 m 56 s .96 +0 S .998 



t 


f 


— 2 m 30 9 


6.2 


2 


4.0 


1 40 


2.8 


22 


0.1 





0.0 


+ 25 


0.2 


1 30 


2.2 


1 55 


3.7 



j 4 .99 
4-5.46 1 .47 
+ 8 2.42 
Watch t. of ap. h h 7 m 30 s . 1 

Hourly ch. — 8 .7 

0's dec. —21° 54' 43".3 + 22".90 



+ 2 5 .3 
—21 52 38 




2 


15 


5.1 


4 


32 


20.6 


4 


55 


24.2 


5 


15 


27.6 



158 NAVIGATION. 



O 43 34 21 Ciln. cor. + 10".l 

< Ref. —58 .5 

— 52 ( Par. + 6 .3 

1".9635 log 0.2930 

96.7 h = 43 33 29 Z= + 24° 34' 1. cos 9.9588 

A h= 8.06 x 2".28 = +18 =—21 52 1. cos 9.9657 

h = 43 33 47 X— fc+46 27 1. cosec 0.1398 

z =+46 26 13 J h= 2".28 log 0.3573 

d—— 21 52 38 

Custom-House, X= + 24 33 35 Light-House, X=+ 24° 32' 50" 

2. 1843, January 31 (civil date). Circum-meridian alti- 
tudes of © observed at E. Base station, Mullet Key in 
Tampa Bay; lat. 27° 37' 1ST., long 5 h 30 m 50 s W. 



T. by Chron. 


Sextant JVb. 2. 


Art. hor. No. 1 


h m s 


O i II 




5 30 


2© 90 20 20 A end toward obser. 


6 10 


21 30 




6 45 


22 20 




8 42 


2 89 19 50 




9 12 


20 10 




9 38 


20 40 




11 00 


21 36 B end toward obser. 


11 30 


22 10 




11 52 


22 30 


Index cor. + 1 


13 15 


2 © 90 27 50 




13 45 


28 10 


Chron. cor. — 


14 17 


28 20 




15 25 


28 30 


Bar. 29.95 


15 50 


28 40 




16 18 


28 40 


Ther. 72° 


17 53 


2 89 23 




18 20 


22 50 




18 40 


22 50 




20 50 


21 A end toward obser. 


21 12 


20 40 





LATITUDE BY CIRCUM-MERIDIAN ALTITUDES. 159 



T 


by Chron. 

h . m a 

21 37 




Sextant No. 2. 


Art. hor. No. 1 










o / // 

89 20 30 












22 50 




2 © 90 24 10 












23 20 




23 40 












23 40 




23 30 










Bq. of Time + 13 m 44 s 


19+0 S .374 


1".9635 


log 0.2930 






+ 2 9 . 


07 r 1.87 


X = + 27° 37' 


1. 


cos 9.9475 




+ 13 m 46 s . 


3 •] .19 


d =—17° 25' 


1. 


cos 9.9796 


— Chron. cor. + 


l m 39 s . 


3 ( .01 L— d = +45° 2' 1. 


cosec 0.1503 


Chr. T. ap. Noon + 15 m 26 8 . 


A 


Q h = 2" 


346 


lo 


g 0.3704 




t 

m s 

—9 56 


98.7 


(k) 

O I II 

45 10 10 


2".346 t* 

i a 

+ 3 51 


S. diam 
/ // 
—16 16 




(K) 

o ; // 
44 57 45 




9 16 


85.9 


10 45 


3 21 






50 




8 41 


75.4 


11 10 


2 57 






51 




6 44 


45.3 


44 39 55 


1 46 


+ 16 16 




57 




6 14 


38.9 


40 5 


1 31 






52 




5 48 


33.6 


40 20 


1 19 






55 




4 26 


19.7 


40 45 


46 






47 




3 56 


15.5 


44 41 5 


36 






57 




3 34 


12.7 


41 15 


30 






61 




2 11 


4.8 


45 13 55 


11 


—16 16 




50 




1 41 


2.8 


14 5 


7 






56 




1 9 


1.3 


14 10 


2 






56 




—0 1 


0.0 


14 15 









59 




+0 24 


0.2 


14 20 









64 




52 


0.8 


14 20 


2 






66 




2 27 


6.0 


44 41 30 


14 


+ 16 16 




60 




2 54 


8.4 


41 25 


20 






61 




3 14 


10.5 


41 25 


25 






66 




5 24 


29.2 


40 30 


1 8 






54 




5 46 


33.3 


40 20 


1 18 






54 




6 11 


'38.2 


40 15 


1 30 






61 




7 24 


54.8 


45 12 5 


2 9 


—16 16 




58 




7 54 


62.4 


11 50 


2 26 






70 




+ 8 14 


67.8 


11 45 


+2 39 






68 


Mean 


—0 32 






Mean 0's U. L. 


44 57 58 


Long. 


5 h 30 50 








O'sL. 


L 


57 


G. ap. 


T. 5 30 18 








A 
B 




44 57 57 
58 



160 NAVIGATION". 

0\s Dec. — 1°7 28 26.0 + 41.55 h' = 44 57 57.4 

+ 3 48.7^207.75 -Jin. cor. +32.5 

d-__17 24 37.3 j 20.71 Ref. —56.2 

z Q = +45 2 20 ( - 21 Par. +6.2 

Lat. - + 27 37 43 h = 44 57 40 

3. 1865, May 22, 9 h , circum-meridian altitudes of a Vir- 
ginis (Sjiica) at Light-House on St. George's Island, Apala- 
chicola Bay, Florida, lat. 29° 37' K, long. 85° 5' 15" W. 



In. cor. — 3' 0" 
Bar. 30.04, Ther. 73° 

Chro. cor. (L.m.t.)+5 h 35 m 32 9 .9 
Long. +5 40 21 



T. by Chro. 


Sextant No. 1. 


Art. Hor. No. 1 


h m s 


O 1 II 




3 28 56 


2 alt. 99 43 50 


A. end 


31 24 


47 50 




33 36 


51 




34 56 


53 40 




37 8 


53 40 




38 58 


55 30 


B. end 


42 45 


55 30 




44 33 


52 10 




48 21 


46 50 




51 25 


43 50 

99 50 23 





7i'= 49 55 12 ( i In. cor. — 1' 30" 
_2i7( Ref. - 47 
7i = 49 52 55 





h 


m 


3 


*'sR. A. 


13 


18 


7.8 


So 


—4 





30.0 


— Red for % 






-55.9 


Sid. int. from h 


9 


16 


41.9 


Red. 




-1 


31.2 


L. m. t. of transit 


9 


15 


10.7 


— Chro. cor. 


— 5 


35 


32.9 


Chro. t. of transit 


3 


39 


38 



{Mean) (Sid.) 



-10 42 


— 10 44 


115.2 


1".9635 




log 


0.2930 


8 14 


8 15 


68.1 


Z = + 29°37' 




1. cos 


9.9391 


6 2 


6 3 


36.6 


d=r — 10 28 




1. COS 


9.9927 


4 42 


4 43 


22.2 


Z-eJrrr+40 5 




l.cosec 0.1912 


2 30 


2 30 


6.2 


AJi == 2".606 




log 


0.4160 


40 


40 


0.4 


P= 49.86 




log 


1.6979 


+3 7 


+ 3 7 


9.7 


Ah - + 2' 


10" 


log 


2.1139 


4 55 


4 56 


24.3 


h — 49° 52 


55 






8 43 


8 44 


76.3 


h Q = 49 55 


5 






11 47 


11 49 


139.6 

49.86 


* =+40 4 
tf=-.10 27 

X=r + 29 37 


55 
34 
21 







LATITUDE BY CIRCUM-MERIDIAN" ALTITUDES. 161 



160. Problem 48. To find the latitude from an observed 
altitude of Polaris or the North Pole-star. 
Solution. The formulas (112) of Prob. 46, 

tan ■==. cot d cos t 
cos sin A 



cos (/)'- 



sin d 



90°— Z = ±4>' 
can be greatly simplified in the case of the Pole-star, since 
its polar distance is only 1° 25'. 

Putting 
we have 



d— 90°— p and 0'= 90°— cj>% 

tan <p = tan^ cos £ 



or, 



<p =p cos £ (within 0".5) > 

COS 



sin </>"= sin A 
i 



r 



(125) 



(126) 



cosp 

■0'-0, J 

the 2d value of i, or (180°— <//'— 0), being excluded, as it 

exceeds 90°. p and are so small, that the cosine of each 

is nearly 1, and consequently 

sin $"= sin h and 0"= h, nearly. 

Thus we have 

r=p cos t 

Z = h-<j> 
If t is more than 6 h or less than 18 h , cos t is negative, and 

we have numerically 

L = h + 0. 

Let $ represent the sidereal time, and a the right ascen- 
sion of the star, then 

t = S—a and =p cos (S —a). 

If we consider the right ascension and polar distance of 
the star to be constant, may be computed and tabulated 
for different sidereal times (right ascensions of the meridian), 
as in Bowditch, p. 206, and "Tab. I. for the Pole-star" in 



162 



NAVIGATION. 



the British Nautical Almanac. But owing to the change 
of right ascension and declination, such a table requires cor- 
rection for each year. It is better to take the apparent right 
ascension and declination from the Almanac, and compute 
t and <p. 

(j> may be found approximately in the traverse table (Tab. 
II.) in the Lat. col., by entering the table with t as a course, 
and p as a distance. 

161. Formulas (126) maybe de- 
rived from Fig. 34, by regarding 
P M m as a plane triangle, and 
Z m = Z M. The first produces 
no error greater than 0".5. The 
error of the second is evidently 
greater the greater the altitude, or 
the latitude. This error, however, 
will not be more than 0'.5 in lati- 
tudes less than 20°, nor more than 
2' in latitudes less than 60°. 

162. We may use (125) with more exactness, but these 
formulas may be modified so as to facilitate computation. 
Put <b rr =h+Ah 

then, changing the 2d of (125) to a logarithmic form, we 
have 

log sin (h+A h) = log sin h + log cos <£ — log cosp> 
or 

log sin (h+A h) — log sin h = log seep — log sec 0. 

But A h being very small, representing by D ;/ the change of 
log sin h for 1", we have, with J A in seconds, 

log sin (h +Ah)— log sin h — A h X D M ; 
whence, by substituting in the preceding, we obtain 
log sec p — log sec (f> _ log cos <j> — log cos p 




Jh = 



D m 



D. 



(127) 



LATITUDE BY CIRCUM-MERIDIAN ALTITUDES. 163 



The difference of the log secants, or log cosines, of p and $ 
is readily taken from the table by inspection. D ;/ for log sin# 
is usually given in tables of 7 decimal places, and hence 
A h is readily found. 

We have then 

0==pOOfl.t \ (12S) 



If D, is the change of log sin h for 1', then in minutes 
log sec p — log sec <j> 



Ah = - 



D, 



(129) 



163. The British Nautical Almanac contains three tables 
for the reduction of altitudes of Polaris, from which they 
may be found to the nearest second. 

164. Altitudes of Polaris may often be observed at sea, 
with some degree of precision, during twilight, when the 
horizon is well defined, and the latitude found from them 
within 3' or 4'. 

Examples. (Prob. 48.) 

1. At sea, 1865, March 31, 7 h 15 m 19 s , mean time in long. 
160° 15' E. ; obs'd alt. of Polaris 38° 18' ; index cor. +3'; 
height of eye 17 feet : what is the latitude ? (128) 



h m s 

L. m. t. March 31 7 15 19 Long. 


h m s 

—10 41 




3, 


35 29 






Red. for long. 


— 145 






Red. of L. m. t. 
L. sid. t. 
*'sR. A. 


+ 1 11 K— 38° 18' / 
7 50 14 —2 - 
1 9 14/ir: 38 16 ( 


In. cor. +3 
Dip -4 
Ref. —1 




i- 


= 6 41 t— 100° 15' 


1. cos 9.250 n 






— p=— 1 24.5 
— ^r=+ 15.0 


log 1.927/1 
log 1.177 


1. sec .00013 
1. sec 




Jh=+ 0.8 
h— 38 16 


1. sec p — 1. sec d> 


16 




V, 






L=+ 38 32 







164 NAVIGATION". 

2. 1865, May 22, 9 h ; altitudes of Polaris, at light-house 
on St. George's Island, Apalachicola Bay: lat. 29° 37' N". ; 
long. 85° 5' 15" W. ; sextant No. 1, index cor. ~3'0"; 
Art. Hor. No. 1 ; Bar. 30.04, Ther. 73° ; Chro. cor..(L. m. t.) 
+ 5 h 35 m 33 9 . 

KlyCkro. iA JXfhor.) T.ly Chro. ^dofHor.) 





3 IT 25 


56 32 20 






4 2 18 


56 34 




19 26 


32 30 






6 3 


34 20 




21 39 


32 50 






8 27 


34 50 




30 35 


32 80 






10 15 


34 50 




3 22 16 




T. by Chro. 


14 40 
4 .8 21 


85 40 


T. by Chro. 


27^=56 32 32 


27i'=56 34 44 


Chro. cor. 


+ 5 35 33 


7*/= 28 16 16 


Chrc 


. cor. 


+ 5 35 33 


7i/=28 17 22 


L. m. t. 


8 57 49 


i-In. cor.-l 30 


L. m 


. t. 


9 43 54 


Jin. cor.— 1 30 


#0 


4 30 


Ref. -1 43 


So 




4 30 


Ref. -1 43 


Bed. for 2, 


+ 56 


h=2S 13 3 


Red. 


for 2, 


+ 56 


7t=28 14 9 


Red. for m. t. 


+ 1 28 




Red. 


for m. t. 


1 36 




L. sid. t. 


13 43 




L. sid. t. 


13 46 56 




* 's It. A. 


1 9 82 




^'s 


R. A. 


1 9 82 




H 


11 51 11 
177° 47' 45" 


1. cos 9.9996S n 




H 


12 37 24 
189 c 21' 0" 


I. cos 9.99419 n 


-p=- 


1 24 44 


log 3.70621 n 




— p=— 


1 24 44 


log 3.70621 n 


-0= + 


1 24 40 


log 3.70599 




-$= + 


1 23 37 


log 3.70040 


A*<= 





l.secp 1319 \ 

1. sec<£ 1317 [• 

(Zf,=39.2) 2 J 




A*<= + 


1 


l.secp 1319 I 
I. sec <j> 1284 y 


7i = 


28 13 3 




h= 


28 14 9 


z= + 


29 37 43 




L= + 


29 37 47 


CZf„=39.2) 35 J 



1. sec^> and 1. sec (f> are expressed in units of the 7th place 
of decimals. 



CHAPTER VIII. 
THE CHRONOMETER,— LONGITUDE. 

165. Astronomically the longitude of a place is the dif- 
ference of the local and Greenwich times of the same instant. 
It is west or east, according as the Greenwich time is greater 
or less than the local time. (Art. 73.) 

The mean solar, the apparent, or the sidereal times of the 
two places may be thus compared. 

166. A chronometer is simply a correct time-measurer, but 
the name is technically applied to instruments adapted to 
use on board ship. It is here used more generally, as in- 
cluding clocks which are compensated for changes of tem- 
perature. 

A mean time chronometer is one regulated to mean time ; 
that is, so as to gain or lose daily but a few seconds on mean 
time. 

A sidereal chronometer is one regulated to sidereal time. 

167. A chronometer is said to be regulated to the local 
time of any place, when it is known how much it is too fast, 
or too slow, of that local time, and how much it gains or 
loses daily. The first is the error (on local time) ; the 
second is the daily rate. Both are + if the chronometer is 
fast and gaining. 

It is preferable, however, to use the correction of the chro- 
nometer, which is the quantity to be applied to the chrono- 
meter time to reduce it to the true time, and its daily change* 
Both are + when the chronometer is slow and losing. 



166 NAVIGATION. 

They will be designated by c and A c. 

A chronometer is said to be regulated to Greenwich time, 
when its correction on Greenwich time and its daily change 
are known. 

If c is the chro. cor. to reduce to Greenwich time, and e, 
the chro. cor. to reduce to the time of a place whose longi- 
tude is X (-{- if west). 

<? z=c+A, or c = c — A; (130) 

so that the one can readily be converted into the other. 

168. If the correction of the chronometer at a given date, 
and its daily change, are known, the correction at another 
date can easily be found. For let 

c be the given correction at the date T, 
c', the required correction at the date T \ 
t=T'—T, expressed in days, 
A c, the daily change ; 

then c'=c + t. A c. (131) 

t is negative if the date for which the correction is re- 
quired is before that for which it is given. 

If A c is large, t must include the parts of a day in the 
elapsed time. 

A c may be given for two different dates, and vary in 
value. It may then be interpolated for the middle date be- 
tween the two of this problem. 

Thus, if A'c be a second value determined n days after 

the first, the daily variation of A c, regarded as uniform, 

will be 

A'c- Ac 

n (132) 

Representing this by A. z c, we have for the mean daily change 
of the chronometer correction during the period t y or that at 
the middle date, 

A c + \ t. J 3 c, 



THE CHRONOMETER. 167 

and the required chronometer correction, 

c r = c + t. A c + i f. 4 c. (133) 

When the chronometer is in daily use, it is convenient to 
form a table of its correction for each day at a particular 
hour. For a stationary chronometer, the most convenient 
hour is h of local time ; for a Greenwich chronometer, h of 
Greenwich time. 

Examples. 

1. Chro. 16 75, regulated to Greenwich mean time ; 1865, 
Jan. 15, h ; correction + l h l m 25 9 .0 ; daily change — 7 S .65; 
required the correction, Jan. 26, 6 h . 

Jan. 15, h , Chro. cor. +l h 16 m 25 9 .0 

— 7 s .65x 11.25 == — I 26.1 

Jan. 26, 6 h Chro. cor. + 1 14 58 .9 

This chronometer is sloio and gaining. 

2. To find the chro. cor. to reduce to local time, Jan. 26, 
h , in long. 85° 16' E. 

Chro. cor. (Jan. 26 6 h G. t.) +l h 14 m 58 9 .9 

—Long. +6 +5 41 4 

Red. for —12 +3.8 

Chro. cor. (Jan. 26 L. t.) + 6 56 6.7 or — 5 h 3 m 53 9 .3 

3. To form a table of chronometer correction for each 
day from Jan. 26, 6 h to Feb. 6, 6\ 

G. m. t. Chro. cor. 

Feb. 1 6 h + 1* 14 m 13 s .O 

2 6 14 5 .4 

3 6 13 57 .7 

4 6 13 50 .1 

5 6 13 42 .4 

6 6 +1 13 34 .8 

169. To find the rate, or daily change, of a chronometer, 
it is necessary to find the correction of the chronometer on 
two different days, either from observations, or by compari- 



G. 


m. t. 


Chro. cor. 


Jan 


26 6 h 


-f-l h 14 m 58 9 .9 




27 6 


14 51 .3 




28 6 


14 43 .6 




29 6 


14 36 .0 




30 6 


14 28 .3 




31 6 


+ 1 14 20.7 



168 NAVIGATION. 

son with a chronometer, whose correction is known. Let 
c l and c 2 be the two corrections, t the interval expressed in 
days; then we have for the daily change, 

^ = ^~; (134) 

that is, the daily change is equal to the difference of the two 
chronometer corrections divided by the number of days and 
parts in the interval. If attention is paid to the signs, + 
will indicate that the chronometer is losing^ — that it is 
gaining. 

Examples. 



Chro. 1615 


Chro. 4872 


Chro. 796 


h h m s 


h m s 


h m 8 


Chro. cor. April 15 +0 18 16.2 


—1 15 27.5 


+0 16.6 


" " " 27 8 +0 18 29.6 


—1 14 58.6 


—00 5.3 


Change in 12.3 days, +13.4 


+ 28.9 


—21.9 


Daily change of cor. 4-1.09 


+ 2.35 


—2.71 



At fixed observatories an interval of one day may suffice. 
For rating sea-chronometers by observations made with a 
sextant and artificial horizon, an interval of from 5 to 15 
days is desirable. 

The sea-rate of a chronometer is sometimes different from 
its rate on shore, or even from its rate while on board ship 
in port. Some chronometers are affected by magnetic in- 
fluences, so that their rates are varied by changing the di- 
rection of the XII. hour mark to different points of the ho- 
rizon. All are slightly affected by changes of temperature, 
as perfect compensation is rarely attainable. The excellence 
of a chronometer depends upon the permanence of its rate. 
The rate may be large, but if its variations are small, the 
chronometer is good. 

170. A watch is often used for noting the time of an ob- 
servation. It is compared with the chronometer by noting 



THE CHRONOMETER. 169 

the time of each at the same instant. The most favorable 
instant is when the watch shows no s . 

Let C and 77" be these noted times; then A W— ( C — W) 
is the reduction of the watch time to the chronometer time : 
for C=W+(C — W). 

Comparisons should be made before and after the observa- 
tion, and the results interpolated to the time of observation. 

A practised observer may, by looking at the watch and 
counting the beats of the chronometer, make the comparL 
son to the nearest S .25. It is better to take the mean of 
several comparisons than to trust to a single one. 

A mean time and a sidereal chronometer may be com- 
pared within s . 03 by watching for the coincidence of beats, 
which occurs at intervals of 3 m , for chronometers, which 
beat half-seconds. 





Examples. 






Chro. 476 

h m s 

Chro. 4 16 56.2 


Chro. 4072 

h m s 

3 15 17.5 


Chro. 1976 

h m 8 

11 48 18.2 


Chro. 1976 

h m s 

1 28.5 


Watch 15 


7 35 30 


3 16 


4 28 


C7 — TFI-t-3 11 56.2 


—4 20 12.5 


—3 27 41.8 


—3 27 31.5 



The last two are comparisons of the watch with the same 
chronometer. Suppose the time of an observation as noted 
by the watch to be 3 h 3Y m 17 s ; for finding the corresponding 
time by the chronometer we have, 

The change of C— W in 1\2, + 10 s .3 ; 
whence the change in l h is + 8 .6, 

and the change in 21 m .3 = 0\35, the interval between the 
1st comparison and the observation, +3 s .O ; 
or, by proportion, we have 

72 m : 21 ra .3 =: + 10 s .3 : +3 s .O 
Then, Time by Watch = 3 h 37 m 17 s 
C—W — — 3 27 . 38.8 
Time by chro. = 09 38.2 



170 JSTAV1GATION. 

171. Problem 49. To find the correction of a chronome- 
ter at a place whose latitude and longitude are given. 

1st Method. (By single altitudes.) 

Observe an altitude, or set of altitudes, of the sun or a 
star, noting the time by the chronometer, or a watch com- 
pared with it. 

Find from the altitude (Prob. 43) the local mean, or 
sidereal, time, as may be required. 

The " local time " — the " chronometer time," or 

c—T-C 

(art. 135), is the correction of the chronometer on local 
time. Applying to this the known longitude of the place 
of observation, gives the correction on Greenwich time. 

172. If an artificial horizon is used, as it should be when 
practicable, it is best to make two sets of observations with 
the roof in reversed positions. In A. M. observations of the 
sun with a sextant and artificial horizon, the lower limb of 
the sun and the upper limb of its image in the horizon are 
made to lap, and the instant of separation is watched for ; 
while in P. M. observations the limbs are separated and ap- 
proaching, and the instant of contact is noted. In observa- 
tions of the upper limb this is reversed. Even a good ob- 
server may estimate the contact of two disks differently 
when they are separating and when they are approaching. 
Both limbs, then, should be observed. 

In observing altitudes which change rapidly it is better, 
when circumstances permit, to set the 'instrument so as to 
read exact divisions at regular intervals, and watch the in- 
stant of contact. A good observer, with a sextant and ar- 
tificial horizon, can observe the double altitudes at regular 
intervals of 10'. 

1 73. On a subsequent day repeat this observation, and 
find ai^ain the correction of the chronometer. The differ- 



THE -CHRONOMETER. - 171 

ence between these two corrections divided by the number 
of days and parts in the interval is the daily change, as in 
Art. 169. 

It is important that both the observations thus compared 
should be at nearly the same altitude and on the same side 
of the meridian (when the sun is observed, both in the fore- 
noon, or both in the afternoon), and in general, that they 
should be made with the same instruments, and as nearly as 
practicable under the same circumstances. Thus, an error in 
the assumed latitude and constant errors of the instruments 
or the observer w T ill affect the two chronometer corrections 
nearly alike, but will very slightly affect their difference, 
and, consequently, the rate determined from it will be nearly 
exact. The chronometer correction, derived from single al- 
titudes, may be erroneous a few seconds. But for sea 
chronometers this is of less importance than an erroneous 
determination of the rate. For instance, suppose the deter- 
mined chronometer correction in error 4 s , and the daily 
change in error I s ; in 20 days (Art. 168) the computed 
change of the correction will be in error 20 s , and in 30 days 
will be in error 30 s . 

174. 2d Method. (By double altitudes.) 

It is better to observe altitudes of the body on both sides 
of the meridian, and as nearly at the same altitude as prac- 
ticable, either on the same day or on two consecutive clays. 

Altitudes of two stars also may be used, one east, the 
other west of the meridian. 

The mean of the two results is better than a determination 
from either alone ; for constant errors of the latitude, the 
instrument, or the observer, affect the two results in oppo- 
site directions ; that is, if one result is too large, the other 
is too small, and by nearly the same amount. 



172 . NAVIGATION. 

Examples. (Prob. 49.) 

1. Chronometer Correction. 

Pensacola Navy- Yard, 30° 20' 30" K, 87° 15' 21" W. 
1865, May 30 21^; Chro. 1876. 



T. by Chro, 


Sextant No. 2. 


Art. Hor. No. 1. 


m s 




o / 


m s 


31 41 


3 


2 © 99- 50 A. end. 


Chro. cor. (G. m. t.)— 42 26 




22.7 




Daily change — 3. 


32 3.7 




100 






23.3 




O's diam. off arc +32 8.3 


32 27 


24 


100 10 


on arc— 30 59.2 


32 51 


23 


100 20 


In. cor. +- 34.5 


33 14 


23.7 


100 30 




33 37.7 




100 40 




34 7.5 


23 


2 99 50 B. end. 


Bar. 30.14 


34 30.5 


23 


100 


Ther. 76° 


34 53.5 


23.3 


100 10 




35 16.8 


23 


100 20 




35 39.8 


23.2 


100 30 




36 3 




100 40 





4 



3 32 39.07 2 © 100 15 

3 35 5.18 2 100 15 



Computation* 



h 



T. by Chro. 3 32 39.07 O V dec. Eq'n of t. 

o I II ii m s 8 

Chro. cor. -42 26 +2157 40.3 +21.04 +2 36.35 —0.360 

.720 



f 42.08 f .72 

+ 59.7 I 16.83 —1.02 I .28 



G.m. t. May 31 2 50 11 

2.837 +2158 40.0 .15 +2 35.33 ^ 2 



1 .62 \ 

>.0 [ .15 +2 35.33 [ 



THE CHRONOMETER. 



173 



h 

30 21 



50 7 30 
— 16 17.3 
h = 49 51 12.fi 

L = 30 20 30 
p = 68 1 20.0 



L. ap. t. May 30 21 3 51.35 
— Eq. of t, — 2 35.33 

L. m. t. May 30 21 1 16.02 

T. by Chro. 3 32 39.07 2 s = 148 13 2.7 

0,Chro. cor.(L.m.t)— 6 31 23.05 s = 74 6 31.4 

is— 7i = 24 15 18.7 

O / // 

I * = 157 58 55.1* 
t= 315 57 50.2 



\\n. cor. // a 

+ 17.3 ref. — 51.6 

S. diam. 

— 15' 48.4 par. +5.4 
1. sec 0.0639749 
1. cosec 0.0327661 



1. cos 

1. sin 



9.4374537 

9.6136320 

19.1478267 

9.5739134 



T. by Chro. 3 35 5.18 O's dec. Eq'noft. 

oil, it m s* 

Chro. cor. — 42 26 +21 58 40.0 +21.04 +2 35.33 
G. m. t. May 31 2 52 41 in h .041 + .8 



2.878 



+ 21 58 40.8 



— .02 

+ 2 35.31 



—0.360 



L. ap. t. May 30 21 6 17.89 

— Eq. of t. — 2 35.31 

L. m. t. May 30 21 3 42.58 

T. by Chro. 3 35 5.18 

0,Chro.cor.(L.m.t.) -6 31 3 2.60 



O 50 7 30 
+ 15 19.5 j 
h = 50 22 49.5 I 
L — 30 20 30 
p= 68 - 1 19.2 
2 s = 148 44 38.7 
s= 74 22 19.4 
-h = 23 59 29.9 



Mean —6 31 22.82 \ 

' Red for 3 h .0 — .48 

Chro.cor.(L.m.t.)-6 31 23.30 



*■= 158 17 14.2 

t = 316 34 28.4 



i In. cor. // // 

+ 17.3 ref.— 51.6 

S. diam. 

+ 15' 48.4 par. +5.4 
1. sec 0.0639749 
1. cosec 0.0327668 



1. cos 
1. sin 

1. sin 



9.4303807 

9.6091709 

19.1362933 

9.5681467 



May 31 h 



2. Chronometer Correction. 

Pensacola Navy- Yard, 30° 20' 30 /; N"., 81° 15' 21 ;/ W. 
1865, May 31 3 h . 



* In A. M. observations, \ t may be taken in the 2d quadrant ; or it may 
be taken in the 1st quadrant and marked — . 



174 NAVIGATION. 

T. by Chro. Sextant No. 2. Art. Hor. No. 1. 

h m e o / m s 

9 24 2.7 8 2© 100 40 A. end. Chro. cor. (G. m. t.) —42 27 ) 

22.8 Daily change — 3.8 J 

24 25.5 100 30 

23.0 G's diam. off arc +32 12.5 ) 

24 48.5 100 20 on arc —30 59.2 j 

24.0 

25 12.5 100 10 In. cor. + 36.6 

22.3 
25 34.8 100 

23.4 
25 58.2 99 50 

28 33.5 2 97 40 B. end. Bar. 30.14 

23.5 

28 61 97 30 Ther. 76° 

23.5 

29 20.5 97 20 

22.5 

29 43 97 10 

23.0 

30 6 97 

23.5 
30 29.5 96 50 



9 25 0.37 2 100 15 

9 29 31.58 2 97 15 

Computation. 

h m s 

T. by Chro. 9 25 0.37 O's dec. Eq'n oft. 

oil/ II ma e 

Chro. cor. — 42 21 +21 57 40.3 + 20.92 +2 36.35 —0.362 

( 167.36 ( 2.896 

G. m. t. May 31 8 42 33 +3 2.2 \ 14.64 —3.15 \ .253 



12.5 ( 



8.709 +22 42.5 ( I 9 +2 33.20 ( '3 

o / // 

©50 7 30 filn.cor. u u 

-16 16.3 + 18 - 3 ref - 5LG 

S. diam. 

h— 49 51 13.7 [ +15' 48.4 par. +5.4 

L— SO 20 30 1. sec 0.0639749 

h m 8 p= 67 59 17.5 1. cosec 0.0328703 

L. ap. t. May 31 2 56 12.04 2s = 148 11 1.2 

— Eq. of t. —2 33.20 5= 74 5 30.6 1. cos 9.4379032 

L. m. t. May 31 2 53 38.84 s-h= 24 14 16.9 1. sin 9.6133431 

T. by Chro. 9 25 0.37 ~ ~~ 19.1480915 

0,Chro.cor.(L.m.t.)- 6 31 21.53 \ t— 22 130.3 1. sin 9.5740458 

t= 44 3 0.6 



THE CHRONOMETER. 



175 



T. by Chro. 9 29 31.58 O '« dec. Eq'n of t 

Chro. cor. —42 27 +-22 42.5 +20.92 +2 33.20 — 0.352 

G. m. t. May31 8 47 5 inO\076 +1.6 * — .03 

8.785 +22 44.1 + 2 33.17 

48 37 30 (i In. cor. „ „ 

~ +15 17.8 J ,. +18-3 ref.-o-t.o 

I S. diam. 

h m g h— 48 52 47.8 L +15' 48.4 par. +5.6 

L. ap. t. May 31 3 42.95 L — 30 20 30 1. sec 0.0639749 

— Eq. of t. —2 33.17 p = 67 59 15.9 1. cosec 0.0328717 

L. m. t. May 31 2 58 9.78 2s =147 12 33.7 

T. by Chro. 9 29 31.88 s = 73 36 16.8 1. cos 9.4506544 

0, Chro. cor.(L.Di.t.) -6 31 21.80 s-h = 24 43 29.0 L sin 9.6214453 

19.1689463 
(mean) —6 31 21.66 \t — 22 35 22.1 1. sin 9.5844732 

Red for —3' . I +.48 t = 45 10 44.2 

Chro.cor.(L.ni.t.)- 6 31 21.18, May SI O h . 

May 31 h Chro. cor. (L. m. t.) — 6 h 31 m 22 s .24 by A.M. and P.M. obs. 

Long. +5 49 1 .4 

May 31 6 h Chro. cor. (G. m. t.) — 42 20 .84 



3. Table of Chro. Corrections. 
Chro. 1876 ; fast of Greenwich mean time and gaining. 

Remarks. 



O , A.M. Key West Light-House. 
©,A.M. " " " " 

, A.M. & P.M. Pensacola Navy-Yard. 
©,A.M. &P.M. " " " 



G. m. t. 


Chro. cor. 


Daily Ch. 


h 


' h m s 




1865, May 1 3 


— 40 20.5 


-4.14 


17 3 


41 26.8 


3.88 


25 6 


41 58.3 


3.75 


31 6 


42 20.8 





Long* of Key West Light-House, 81° 48' 40" W. 
Long, oi Pensacola Navy- Yard, 87 15 21 W. 

* The assumed longitudes of places, where the chronometer is rated, should 
be stated. 



176 NAVIGATION. 

4. Comparisons and Corrections of Chronometers. 

1865, May 31, 6 h , G. mean time. 

Chro. 4375 Chro. 9163 Chro. 789 Cliro. 5165 

hms hms hms hms 

Chro. 6 50 16.3 5 3 29.7 2 15 27.5 11 59 16.8 

(1876) 6 30 6 31 6 32 10 6 33 30 

(1876)— Chro. —0 20 16.3 +127 30.3 +4 16 42.5 —5 25 46.8 
Cor. of (1876) —42 20.8 —42 20.8 —42 20.8 —42 20.8 

Chro. cor. — 1 2 37.1 —0 45 9.5 +3 34 21.7 —6 18 7.6 

or +5 41 52.4 

175. 3d Method. (By equal altitudes.) 

A heavenly body, which does not change its declination, 
is at the same altitude east and west of the meridian at the 
same interval of time from its meridian passage. 

If, then, such equal altitudes are observed and the times 
noted by the chronometer, or by a watch and reduced to 
the chronometer (Art. 170), the mean of these times, or the 
middle time, is the chronometer time of the star's meridian 
transit. 

The corresponding sidereal time is the star's right ascen- 
sion, when the first observation is east of the meridian ; 12 h -f 
the right ascension when the first observation is west of the 
meridian. 

This, for a mean time chronometer, may be converted into 
local mean time (Prob. 32) ; and for a Greenwich chrono- 
meter into the corresponding Greenwich time. 

Subtracting the chronometer time, we have the correction 
of the chronometer. 

Example. 

1865, Jan. 14, at Washington, in longitude 77° 2' 48' W., 
equal altitudes of a Canis Minoris were observed, and the 
times noted by a chronometer regulated to Greenwich mean 
time ; from which were obtained : 



THE CHRONOMETER. 177 



Mean of chro. times ( % east) 


2 h 16 n 


35 3 .65 


" "' " " (#- west) 


7 59 


16 .38 


Chro. time of >fc's transit 


5 7 


56 .01 


L. sid. t.= %'s R. A. 


7 32 


16 .26 


Long. 


-ho 8 


11 .2 


G. sid. t. 


12 40 


27.46 


■JS (Jan. 14)- 


-19 35 


51 11 


Sid. int. from Jan. 14 h 


17 4 


36 .35 


Red. to m. t. int. 


— 2 


47 .86 


G. mean time Jan. 14 


. 17 1 


48 .49 


Chro. time 


17 7 


56 .01 


Chro. cor. 


— 6 


7 .52 



176. If equal altitudes of the sun are observed in the fore- 
noon and afternoon of the same day, the mean of the noted 
times would be the chronometer time of apparent noon, 
were it not for the change of the sun's declination between 
the observations. 

Problem 50. In equal altitudes of the sun, to find the 
correction of the middle time for the change of the surfs 
declination in the interval between the observations. 

Solution. Let 

h = the sun's true altitude at each observation, 
t — half the elapsed apparent time between the observa- 
tions, 
y o — the mean of the chronometer times of the two ob- 
servations, or the middle chronometer time, 
AT = the correction of this mean to reduce to the chrono- 
meter time of apparent noon ; 
L = the latitude of the place, 
d = the sun's declination at local apparent noon, 
A d = the change of this declination in the time t ; 
then, when both observations are on the same day, 
t + A T Q will be numerically the hour-angle at the A. M. ob- 
servation, 
t—AT 01 the hour-angle at the P. M. observation, 



178 NAVIGATION. 

d— A d) the declination* at the A. M. observation, 
d + A d, the declination* at the P. M. observation. 

By (116), Ave have for the two observations, 
sin A=sin L sin (d— A d)-\-cos L cos (d—Ad) cos (t+ AT ) ) / lq _\ 
sinh=smLsm(d+ Ad) + cos L cos (d+ Ad) cos (t—' A T Q ) ) ^ ' 

But 

sin (d ± A d) = sin d cos A d do cos d sin A d, 
cos (d ±A d) = cos J cos A d =F sin c? sin J J, 
cos (d ±AT )= cos £ cos A T =F sin £ sin J T^. 
Since J c?, and therefore ^ ^, are very small, we may put 
cos A d = 1, sin A d = A d. sin 1", 

cos A T = 1, sin J 7;= 15 AT . sin l" ; 

A d being expressed in seconds of arc, and 
AT Q in seconds of time ; we shall then have 

sin (d ± Ad) = sin d ± A d. sin V cos c?, 

cos (d ±A d) = cos d =F A d. sin l' 7 sin c?, 

cos ( £ ± zf T^) = cos £=F 15 AT . sin l" sin £. 

Substituting these in the two equations (135), subtracting 

the first from the second, and dividing by 2 sin 1", we shall 

have 

=:A d. sin L cos d — Ad. cos L sin d cos £ 

+ 15 J T . cos i cos c? sin t. 

Transposing and dividing by the coefficient of AT^ we find 

the formula 

AT ,_ A d. tan L Ad. tan d . , 

Jio -~T5"7Tn"i" + ~T5la~nT' < 136 ' 

which is called the equation of equal altitudes. 

Let 
A h d = the hourly change of declination at the instant of 
apparent noon, and express 
£, which is half the elapsed apparent time, in hours, 

* Strictly, in the one case, A d should be the change of declination in the 
time t+ AT Q ; in the other, the change in the time t — AT Q . 



THE CHROMOMETER. 179 



then A d = A h d. t, 

and (136) becomes 



A rn d h d. t tan L A h d. t tan d /-, oH \ 

15 sin f T 15tan« v J 



If we put 



*=Tk7^1 ( 138 ) 



15 sin £ 15 tan £ 

and » 

(7 = the chronometer time of apparent noon, we have 

A T — A. A h d. tan L + B. J h c?. tan J ) , _ * 

In these formulas, L and d are + when north, A d and zl h d 
are + when the sun is moving toward the north. 

The coefficient A is — , since t < 12 h , 
" " i? is + when t < 6 h , — when £ > 6 h . 

The computation of the .two parts of A T is facilitated by 
tables of log A and log B. Such tables are given in Chau- 
venet's " Method of finding the error and rate of a chrono- 
meter," in the American Ephemeris and Nautical Almanac 
for 1856, and reprinted in a pamphlet with his "New 
method of correcting Lunar distances." 

The argument of these tables is 2 t, or the elapsed time. 
The signs of A and B are given. 

Apply the two parts of A T , according to their signs, to 
the Middle Chronometer Time ; the result is the Chronome- 
ter Time of Apparent Noon. > 

Apply to this the equation of time (cidding, when the 
equation of time is additive, to mean time ; otherwise sub- 
tracting) ; the result is the Chronometer Time of Mean 
JVoon at the place. 

Applying to this the longitude (in time), subtracting if 
west, adding if east, gives the Chronometer Time of Mean 
JS T oon at Greenwich. 



180 NAVIGATION. 

*12 h — Chro. T. at local Mean Noon, will be the Chro. corvee- 
lion if the chronometer is regulated to local time. 

*12 h — Chro. T. at Greenwich Mean Noon, will be the Chro. 
correction, if the chronometer is regulated to Greenwich 
time. 

177. If a set of altitudes is observed in the afternoon of 
one day, and a set of equal altitudes in the forenoon of the 
next day, the middle time would correspond nearly to the 
instant of apparent midnight ; and half the elapsed time t, 
would be nearly the hour-angle from the lower branch of 
the meridian, or the supplement of the proper hour-angle. 

In this case 

1 80° — (t + A T ) will be the hour-angle at the P.M. observation. 

180°-(t-AT { j) " " " " " " A.M. " 

d— A d, the declination at the P.M. " 

d+Ad, " " " " A.M. " 

and we have for the two observations, as in (135) 

sin h=:sm L sin (d — A d)— cos L cos (d — A d) cos (t + AT ) ) , , ± v 
sin h=smL sin (d + A d) — cos L cos (d-\-A d) cos (t — A T ) ) ^ * ' 

Treating these in the same way as (135) we shall have 

= A d. sin L cos d+ A d. cos L sin d cos t 
— 15 A T cos L cos d sin t ; 
whence 

* This is better noted as h . 
\ These may be written 

— sin h= — sin L sin (d — A d)+cos L cos (d — A d) cos (t-\- A T Q ) 
— sin h = — sin L sin (d-\- J d)-hcos L cos (d -f- J d) cos {t—A T^). 

They differ from (130) in the signs of h and L and in reckoning the hour- 
angles from the lower, instead of the upper, branch of the meridian. This 
would be the case, if we suppose the observations to be referred to the lati- 
tude and meridian of the antipode. The only effect in (130) is to change 
the sign of tan Z, or of the first term in the equation of equal altitudes. 



THE CHRONOMETER. - 181 

A d. tan L A d. tan d 



A T = 

15 sin t 15 tan t 

or, putting as before A d = A h d. t 

. a=- *:■ ' b = 



15 sin t 15 tan t 

A T ——A. J h d. tan Z + JB. A h d. tan <?, (141) 

which differs from (139) only in the sign of A. This is the 
reduction of the middle time to the Chro. Time of appar- 
ent midnight : applying the equation of time reduces it to 
the Chro. Time of mean midnight. 

178. c?, A' d, and the equation of time are to be taken 
from the Almanac for the instant of apparent noon, or of 
apparent midnight, according as the observations are made 
on the same da) T , or on consecutive days. 

2 t is properly the elapsed apparent time. The elapsed 

time by chronometer requires, then, not only a correction 

for the rate, which is 

2 t 

■^—A c, ( -f- when the chronometer loses) ; 

but also a reduction to an apparent time interval, which, 
for a mean time chronometer, is the change* of the equation 
of time in the time, 2 £, additive when the equation of time 
is additive to mean time and increasing, or subtractive from 
mean time and decreasing. For a sidereal chronometer, it 
is the change in the sun's right ascension in the time 2 £, and 
subtractive. 

179. Equal altitudes of the moon or a planet may be ob- 
served ; but in the case of the moon admit of less precision 
than of the sun, and moreover require correction for the 
inequality produced by change of parallax. 

If 2 A a is the increase of right ascension in the interval, 

* The maximum daily change is 30 s . The elapsed time by Chronometer 
is usually regarded as sufficiently accurate. 



182 NAVIGATION. 

the body will arrive at its second position later than would 
a fixed star, supposed coincident with it at the first posi- 
tion ; and the elapsed sidereal time will be greater than the 
double hour-angle of the body by the quantity 2 A a. If 
2 s = the elapsed sidereal time, then in (137) we must take 

2 t — 2 5—2 A a, or t == s— A a. (142) 

If t m = half the elapsed mean time (expressed in hours 

when used as a coefficient), and 
A h a = the increase of right ascension in l h of mean time, 

by (87) s = t m + 9 s .85$5 t m 

and t == t m + t m (9 S .8565 — A h a), (143) 

by which t and 2 £ may be found from 2 t m the elapsed mean 
time. 

In this expression the last two terms are in seconds. Re- 
ducing to hours we have 



3600 J m v ' 3600/ 

If A h d = the change of declination in l h of mean time, then 
in (136) 

A d = t m . A h d 

or, substituting for t m its value from (144), 

4 a 



Ad=t. A h d~ (1,002738 — 



3600. 



Equations (138) and (139) may then be used for other 
bodies than the sun, provided we give t its proper value 
from (142) or (143), and for A h d substitute 

A' h d=A h d+ (1.002738 — 1|^), 

or, which will be sufficiently exact, 

At i at, A^ — 9 9 .856 A , n . . 

A' h d= A h d + — Q . A h d (145) 



THE CHRONOMETER. 183 

180. Observing the double altitudes at regular intervals 
of 10', or 20', especially facilitates the method of equal alti- 
tudes ; for, if the first set is observed at equal intervals, in 
the second the observer, having set the instrument for the 
last reading of the first and observed the contact, for the 
subsequent observations, has only to move back successively 
the same intervals. 

181. It is not requisite that the instrument should give 
the true altitude ; it is sufficient if the altitude is the same 
at the two corresponding observations. Hence the two ob- 
servations should be made with the same instruments, with- 
out change of adjustment, and in general as nearly as prac- 
ticable under the same circumstances. 

This purpose is promoted by making the final movement 
of the tangent screw in both sets always in the same direc- 
tion. Thus, in reversing the movement, the screw may be 
turned a little too far, and then the final contact made by a 
motion in the same direction as before. 

If the sun is used, both limbs should be observed. 

The error arising from want of parallelism of the surfaces 
of the roof-glasses of the horizon is eliminated by having 
the same end of the roof toward the observer. The roof 
may be tested by observing sets of altitudes with it in re- 
versed positions. 

182. Although the readings of the instrument maybe the 
same in the two sets of observations, the altitudes may be 
slightly different, 1st, from changes in the instrument in the 
interval; 2d, from difference of refraction at the two times. 

A change in the index correction may be detected by ob- 
servation ; but there may be expansion or contraction of 
various parts of the instrument which may affect the read- 
ings of the altitudes without altering the index correction. 

The change of refraction may be found by noting the ba- 



184 NAVIGATION". 

rometer and thermometer at each set, and finding the refrac- 
tion for both sets of altitudes. 

183. To correct the middle time for any small difference 
of the altitudes, whether from refraction or actual change 
of readings, Ave may find, from the difference between two 
readings and the difference of the corresponding times, the 
change of time for a change of 1', or 1", of altitude. This 
multiplied by half the inequality of altitudes, expressed in 
minutes, or seconds, will give the correction of the middle 
time, to be added when the P. M. altitude is the greater ; to 
be subtracted when the P. M. altitude is the less. 

If twice the altitude is observed with an artificial horizon, 
we may find the change of time for a change of 1', or \\ of 
the double altitude, and multiply it by the whole inequality 
of the altitudes. 



Examples. (Prob. 50.) 

* 

1. 1865, Jan. 10 9± h A. M. and 2^ h P. M. Equal al- 
titudes of © at the Custom-House, Key West, Florida ; 

24° 33' 20" N*., 81° 48' 31" W. chro. 1085 ; 

chro. cor. (G. m. t.) — 42 m 18 s .0; daily change + 8 S .3. 



Sex. JSfo. 1. 




T. by Chro. Mid 


time. 






Art. Hor. No 


2. 


AM. 


P.M. 6h 


17^ 


A.M. 


P.M. 


o / 




\\ m a 


h m s 


B 






2 60 A. end. 


3 39 28. T 


S 55 59.7 


43.2 


Q's dia)n.+S2 / 25".0 


+ 32 / 26".7 


10 




39 58.8 


55 27.0 


42.9 


-32 41 .7 


—32 43 .8 


20 




40 31.5 


54 55.3 


43.4 


In. cor. —8 .3 


—8.3 


30 




41 4.0 


54 22.0 


43.0 






40 




41 35.7 


53 50.3 


43.0 


Bar. 30.22 


30.18 


50 




42 9.0 


53 16.5 


42.8 


Ther. 77° 


80° 


2qG0 




42 53.5 


52 20.7 


42.6 






10 




43 31.3 


51 53.5 


42.4 


Ref. — 1'34".9 


—1' 34".4 


20 




44 3.8 


51 21.0 


42.4 


Diff. of alt. 


J/t = +0.5 


80 




44 37.5 


50 48.0 


42.7 






40 




45 10.0 


50 15.5 


42.8 


For 2 A h = 10', 


A t = 32s.8 


50 




45 48.7 


49 42.0 


42.8 


2 A h = 1", 


At= 0.055 


GO 25 




3 42 34.21 


8 52 5l~46 









THE CHRONOMETER. 185 



h m s 

Elapsed Chro. t, 5 10 1T.25 




Long. + 5h 27m 14s.5 Eq. of t + 7m 56s.96 + 0s.993 


Mid. Chro. t. 6 17 42.83 




5h.455 +5. 45 j 4.990 
Od.227 +8 2. 41 1 45t > 


Red.for Ah, 0«.055 x 0.5 = + .03 




1st part of Eq. —2.89 




0's dec— 21° 54' 43" J h <2 + 22\78 ch. in Id + 1 ".065 


2d " " M —1.98 




+ 2 5 +.24 .213 


Chro. t. of ap. noon 6 IT 3T.99 




—21 52 33 +23.02 21 


— Eq. of time — 8 2.41 


1/ 


= +24°33 / .3 t. tan 9.6593 d =— 21° 52'. 6 1. tan 9.6037 m- 


Chro. t. of m. noon 6 9 35.58 


J h <Z=+23".02 log 1.3621 log 1.3621 


—Long. —5 2T 14.5 




A log 9.4396 n B log 9.3315 


Chro. t. of a.m. noon 42 21.03 




-23.89 log 0.4615 » — ls.98 log 0.2973 n 


Chro. cor. (G. m.t) —42 21.03 


Jan 10 5b.o 



(The elapsed apparent time is 5 h 10 m 12 s .) 

2. 1865, June 19, 4J h P.M., and 20, 7J A.M.; nearly equal 

alt. of O; at Belize, S.E. pass of Mississippi River, 29° 7' 8" N., 



89° 5' 18" 


W., chi 


o. 1085; 


chro. cor. 


(G. m. t,)— 41 m 28 s ; 


daily change +l s 


; sextant No. 2 ; 


art. hor. No. 1 ; (A. 


end toward observer). 






P.M. 


A.M. 


P.M. A.M. 


Sex. Read. 


T. oy Chro. 


T.by Chro. 


Sex. Read. 


In. cor. +4.V.8 +40".4 


20 65 10 


10 55 4S 


2 22 44.5 


23. 65 30 


Bar. 30 .09 30 .12 


65 


56 10.8 


22 21 


65 20 


Ther. 81° 80° 


64 50 


56 84.3 


21 34 


65 


i In. cor. + ' 20".9 + ' 20". 2 


64 40 


56 57.5 


21 11.2 


64 50 


Ref. —1 23 .7 —1 23 .5 

—1 7.8—1 8.3 


20 C5 30 


57 27.5 


20 41.5 


2"© 65 40 


(P.M.)— (A.M.) 


65 20 


57 50.5 


20 18 


65 30 


Dili, ref., &c. + ' 0".5 


65 10 


5S 14.3 


19 54 


65 20 


Diff. obs. alts. — 5 37 .5 


65 


58 37.3 
10 57 12.53 


19 7.5 
2 20 5S.96 


65 

2© 65 16 15 


Ah-— 5 37.0 


« 
2065 5 


For2jA=10', ZJ*=23 S .3 


&= 32 32 30 






ft' =32 38 7. 


5 2^7i= 1', At= 2.33 



G. ap. t., June 19, I7 h 56 m 21 s .2 = 19, I7 h .939 = 19 d .W 



O 's dec. 


Ahd. < 


Ih. in Id . 


Eq. of t. 




+ i°3 26 23 


+ 2.01 


—1.03 


m s 

+ 59.99 


+ 0.543 


+ 29 
+ 23 26 57 


—.77 
+ 1.24 


HI 


+ 9.74 
+ 1 9.73 


f5.43 
J 3.801 
1 .489 
I 21 



186 NAVIGATION. 

h m s h m s 

Middle Chro. t.* 18 39 5.75 Elapsed T. by Chro.* 15 23 46 

Red. for J/*,— 5'. 62 x 2s. 33=— 13.09 .... +26 

1st part of Eq. +.39 Cli. of Eq. of t. — 8 

2d " " " —.13 Elapsed ap. t. 15 24 4 

Chro. t. of ap. 12h 18 38 52.92 L= +29° V.l 1. tan 9.7459 d= +23° 27' 1. tan 9.6373 

— Eq'n of t. —1 9.73 Ahd=+YM log 0.0934 .... log 0.0934 

Chro. t. of mean 12h 18 37 43.19 A log 9.7550 B log 9.3891 a 

—Long. —5 56 21.2 +0 S .39 log 9.5943 -0U3 log 9.1198 n 

Chro. t. of G. mean 12h 12 41 21.99 ' 

Chro. cor. (G. m. t.) —41 21.99 June 19 18h 

184. 4th method of finding the correction of a chrono- 
meter. (By transits.) 

On shore, the most accurate method of finding the correc- 
tion of a chronometer is by noting the times of transit of the 
sun or a star across the threads of a well-adjusted transit in- 
strument. The mean of these times is taken and corrected 
for the errors of the instrument, or reduced to the meridian. 
In the case of the sun, the transits of both limbs may be ob- 
served ; or only one, and the " sidereal time of the semi- 
diameter passing the meridian," found on page I. of each 
month in, the almanac, added for the limb, which transits 
first ; subtracted for the second limb. 

At the instant of a star's transit of the meridian, the right 
ascension of the star is the sidereal time. The instant of 
transit of the sun's centre is apparent noon. 

From either of these, the local sidereal or mean time, as 
may be required, can be found ; and thence the chronometer 
correction by subtracting the chronometer time of transit. 

The moon should not be used for finding the time, when 
precision is required. Stars are preferred to the sun, either 
when transits are observed, or equal altitudes with the arti- 
ficial horizon ; chiefly because many stars may be observed 

* To obtain these, 24 h was added to the P. M. chro. time. Twice the 
reduction of the middle time for the diff. of alts, is to be added to the 
elapsed time when the P. M. observation is last ; subtracted when the P. M. 
observation is first. This may be neglected unless the diff. of altitudes is 
quite large. 



LONGITUDE. 187 

during the same night, and the instrument is not exposed to 
the rays of the sun. 

185. By repeating the transits on a subsequent day, the 
chronometer correction can be again found, and from the 
two corrections, the rate as in Art. 1 69. If the transit instru- 
ment is not well adjusted, or the instrumental corrections 
are imperfectly knowm, the rate of the chronometer can still 
be quite well determined from transits of the same star, or 
the same set of stars, on different days, provided the position 
of the instrument, or its adjustments, have not been disturbed 
in the interval. 

186. A rough substitute for a transit instrument is a ver- 
tical corner of a building, and a position for the eye in its 
meridian. The instant of the appearance or disappearance 
of a star, or a limb of the sun, may be noted by a chrono- 
meter, and the chronometer correction obtained as with a 
transit instrument ; but with much less accuracy, since the 
mode of observing is rough, and the position for the eye can- 
not be adjusted to the meridian with much precision : still 
the rate may be found with tolerable accuracy from the 
transits of the same body on different days. 

LONGITUDE. 

187. To find the longitude of a place by astronomical ob- 
servations, it is generally necessary to determine independ- 
ently the local and Greenwich times of the same instant. 
The difference of these times is the longitude, which is west 
when the Greenwich time is the greater, and east when the 
Greenwich is the less (Art. 165). This is expressed by (72) 

in w T hich T is the Greenwich time, and 

T y the corresponding local time of the same kind* 
These times may be apparent, mean, or sidereal. 




188 NAVIGATION. 

The apparent time is the hour-angle of the true sun ; the 
mean time, that of the mean sun ; the sidereal time, that of 
the vernal equinox. In the same way we may use the local 
and Greenwich hour-angles of any other body or point of the 
heavens, regarded as + toward the west. 

This is evident from Fig. 35 ; for if 
P M is the meridian of Greenwich, 
P M', the local meridian, 
P S, the declination circle of a 

heavenly body ; 
M P M' will be the longitude of 

the place, 
M P S, the hour-angle of the 

body at Greenwich, 
M' P S, the local hour-angle ; 
and we shall have, as in Art. 74, 

M P M'= MPS-M'P S. 

The several methods of finding the longitude differ in the 
modes of finding and comparing the two times, or the two 
hour-angles. 

188. Problem 51. To find the longitude of a place by a 
portable chronometer regulated to Greenwich time. 

Solution. The correction and rate of the chronometer are 
supposed to have been found by suitable observations at a 
place whose longitude is known. Let the chronometer be 
transported to the place whose longitude is required ; and 
let an observation suitable for finding the hour-angle of a 
heavenly body, or the local time, be made and the time 
noted by the chronometer, or by a watch compared with it. 

There are then two parts of the process to be pursued : 
1st, from the noted time to find the Greenwich time (mean, 
apparent, or sidereal), or the hour-angle of the body, as may 
be deemed most convenient. 2d, from the observations, to 
find the corresponding local time, or hour-angle. Subtract- 



LONGITUDE. 189 

ing the local time, or hour-angle, from the Greenwich time, 
or hour-angle, will give the longitude. 

189. 1st. To find the Greenwich time, or -hour-angle, of 
the body observed, apply to the noted time the reduction of 
the watch time to chronometer time, C — W (if a watch 
has been used), and the chronometer correction, c', reduced 
to the date of observation (Art. 168). 

The result is, the Greenwich time ; and will be mean or 
sidereal, according as the chronometer is regulated to mean 
or sidereal time.* 

If it is sidereal time, it will be necessary to reduce it to 
mean time (Prob. 32), except when a fixed star has been ob- 
served, so as to take from the almanac the quantities which 
will be required. * 

If, now, the Greenwich hour-angle of the body observed 
is desired : 

In the case of the sun, reduce the Greenwich mean time 
to apparent time, by applying the equation of time. 

If some other body has been observed, reduce the Green- 
wich mean time to sidereal time by adding the right ascen- 
sion of the mean sun ; and thence find the hour-angle of the 
body, by subtracting its right ascension. Or, if a sidereal 
chronometer has been used, from the Greenwich sidereal 
time subtract the right ascension of the body. 

Attention to the signs will give the hour-angle thus ob- 
tained, -f if toward the to est, — if toward the east. 

190. The Greenwich time or hour-angle is affected by the 
error of the chronometer correction, which consists, 1st, of 
the error in its original determination, which includes any 
error of the assumed longitude of the place of rating ; 2d, of 
the error arising from an erroneous rate. This last error is 
cumulative, increasing with the number of days from the 

* For observations of stars, a sidereal chronometer is most convenient. 



190 NAVIGATION. 

date, when the correction of the chronometer was found 
from observations. 

191. The chronometer correction for the date of observa- 
tion can be derived from subsequent as well as from prior 
determinations of it and its daily change. In finding the 
longitude of a place on shore, or of a shoal, both values 
should be obtained, when practicable, and combined by giv- 
ing weights to each inversely proportional to its interval of 
time from the original determination. Thus, if c' and c" are 
two such chronometer corrections, the first brought forward 
t' days, the second carried back t" days, we may take as the 

mean value* 

t" c'+-f c" 

t'+i" ' 
or, in a form more convenient for computation, 

, t' (c,"- Q 

c+ v+r ' 

For example, suppose that on Jan. 17, the chronometer 
correction brought forward from Jan. 1, is -— 18 m 56 s .5, 
and reduced back from Jan. 25, is — 19 m 3 s . 4 ; the value by 
the above formula will be 

If, v fis Q 

— 18™ 56 s .5 + -— — = — 19 m 1M. 

24 

Two longitudes may be combined in a similar way. 

192. Reports of longitudes by chronometer are regarded 
as of but little value, unless the number of chronometers, 
the assumed longitude of the place, where the chronometer 

* This assumes that c' and c" are derived from two chronometer correc- 
tions of equal weight, and consequently that the longitudes used in finding 
them are equally reliable. This may not be the case if the chronometer 
corrections were found from observations at two different places. 

The student is referred to Chauvenet's Astronomy, Vol. I., pp. 317, &c, 
for the methods of allowing for changes in the rates and combining the 
results of several chronometers. 



LONGITUDE. 191 

is rated, and the age of the rates are stated.* Strictly, the 
chronometer merely determines the difference of longitude 
between the two places, where the observations are made. 
This may be obtained by using the chronometer correction 
on the time of the place of rating, instead of the Greenwich 
time. It is preferable to report such differences rather than 
absolute longitudes. 

193. 2d. To find the hour-angle of the body, and thence 
the local time. 

1st Method. (Problem 43. By single altitudes.) Observe 
in quick succession several altitudes of the heavenly body, 
noting the time of each by the chronometer, or by a w r atch 
compared with it. 

Take the mean of the noted times, and from it find the • 
Greenwich mean time ; for which take from the Almanac 
the declination of the body, its semidiameter and horizontal 
parallax when sensible, as well as the quantities required 
for finding the Greenwich hour-angle. (Art. 189.) 

Take the mean of the readings of the instrument, with 
which the altitudes were measured, and from it find the true 
altitude of the centre of the body (Art. 118). With this 
and the known, or assumed, latitude of the place find the 
local hour-angle of the body by Problem 43. 

This hour-angle, which for the sun is the local apparent 
time, subtracted from the corresponding Greenwich hour- 
angle already found, will give the longitude. 

Or, the local mean time may be found from it, for the sun, 
by applying the equation of time ; for other bodies, by add- 
ing the right ascension of the body, which will give the 
local sidereal time, and subtracting the right ascension of 
the mean sun (Prob. 37) : and the local time subtracted 
from the corresponding Greenwich time wall give the longi- 
tude. 

* See U. S. Xavy Regulations, 459. 



192 NAVIGATION. 

194. On shore, it is best to use an artificial horizon, even 
when a sea-horizon can be had, and for precise observations, 
stars in preference to the sun. 

"At sea, the sun is most conveniently used ; but altitudes 
of the moon and bright stars can be employed, when the 
sun is not available. The chief difficulty is the obscurity of 
the sea-horizon at night. During twilight, however, or in 
a bright moonlight, it is often distinct and well defined. 

195. The most favorable position of the body for finding 
its hour-angle from its altitude is, as previously stated, when 
it is nearest the prime vertical ; provided its altitude is not 
so small, as to involve to too great an extent the uncertainty 
of refraction ; and, observed on shore, is within the limits* 
of the instruments employed. 

On shore, the time and circumstances most favorable for 
observations can generally be selected. At sea, long con- 
tinuance of bad weather may render poor observations, made 
under unfavorable circumstances, the only ones available. 

While, then, it is not well to use for finding the time 
an altitude less than 10°, or of an object, whose azimuth is 
less than 45° or more than 135°, it may sometimes be neces- 
sary to exceed these limits. 

196. "When the declination and latitude are nearly the 
same, the body is nearest the prime vertical but a short time 
before and after its meridian passage, so that a very great 
altitude may be used. Thus in lat. 20° N\, the sun, when its 
declination is 19° 55' N. or 20° 5' N., is nearest the prime 
vertical within 22 m of noon at an altitude of nearly 85° ; 
and the local time can be as accurately obtained from an 
altitude of 89°, 4 m from noon, and about 5° in azimuth from 
the prime vertical, as from an altitude of 30°, provided the 
assumed latitude can be depended on within 2'. Nearer 
noon, the rapid change of the sun's azimuth, averaging 10° 

* For a sextant and artificial horizon, between 20° and 60°. 



LONGITUDE. 193 

in l m , would make it difficult to observe the altitude with 
sufficient precision. 

197. The local time or hour-angle is affected by errors in 
the altitude and in the assumed latitude. (Arts. 136, 138.) 
When several observations have been made in rapid succes- 
sion, the effect of a supposed error of 1' in the altitude* 
may be found by dividing the difference of two of the noted 
times by the difference, in minutes, of the corresponding 
altitudes. 

In a similar way we may find the change of altitude 
in l m of time by dividing the difference of two altitudes by 
the difference in mi?iutes, of the corresponding times. The 
maximum change of altitude in l m is 15' ; when L = and 
d = 0. The more rapid the change of altitude, the less will 
errors of altitude affect the result. 

To ascertain the effect of an error of 1' in the assumed 
latitude,! the local times or hour-angles may be computed 
separately for two latitudes differing 10', or 20', from each 
other, and the difference of these times divided by 10', or 
20'. At sea, the latitude by account is used, either brought 

* From the equations, 

sin h — sin L sin d -f- cos L cos d cos t, 

sin (h+A h) = sin L sin d + cos L cos d cos (t+ A t), 

cos d sin t 



sin Z -. 



cos h 



we shall find by processes similar to those pursued in Art. 176, on the sup- 
position that A h and A t are ver) T small, 

4 4 A II 

A t = > 

15 cos L sin Z 

which is a minimum, when Z = ±90°, and incalculable when Z= or 180°. 

f From (118) we find 

At ——- 



15 cos L tan Z 
which is o, when Z= ±90°, and also incalculable when Z— or 180°. 



194 NAVIGATION. 

forward to the time of observation from a preceding, or car- 
ried back from a subsequent determination. It may be very 
largely in error, especially in uncertain currents, or after 
running several days without observations. 

A small error may also result from the assumption that 
the mean of the instrumental readings corresponds to the 
mean of the noted times. The reduction of the mean of 
the altitudes to the mean of the times can be found,* but 
it can be avoided by limiting the series of observations, 
which are combined together, to so brief a period, that the 
error becomes insensible; or, when the body is near the 
meridian in azimuth, by reducing each observation by it- 
self. This last case, however, should be avoided in this 
problem. 

198. At sea, it is usual to reduce longitudes obtained from 
day observations to noon by allowing for the run of the 
ship in the interval, and for currents when known. Those 
from night observations are recorded for the time of ob- 
servation ; or reduced to the commencement or end of the 
watch. 

199. 2d Method. Altitudes in the forenoon and in the 
afternoon, or on different sides of the meridian, are prefer- 
able to single altitudes for finding the local time, for the 
reasons already stated in Article 174. The longitudes can 
be found from each set separately and then combined. 

At sea the longitudes derived from each can be reduced 
to noon, and the mean of the two taken as the true longi- 
tude ; or, if the difference can be regarded as due to cur- 
rents, the longitude at noon can be found by interpolating 
for the elapsed time. It is desirable that the observations 
should be made at nearly equal intervals from noon. 

Longitudes by A. M. and P. M. observations are enjoined 



Chauvenet's Astronomy, Vol. I., p. 214. 



LONGITUDE. 195 

in the directions of the Navy Department whenever prac- 
ticable. 

Examples. (Prob. 51.) 

1. At sea, May 17, 9 h 45 m A. M. ; 24° 50' K, 82° 18' W. 
by reckoning from preceding noon ; 

T. by Watch 9 h 30 m 15 s ; obs'd altitude of Q_ 58° 17'; 
Chro. — Watch + 5 h 12 m 26 s ; Chro. cor. + 25 m 15 s ; 
Index cor. of sextant -f- 3' 20"; height of eye 18 feet; re- 
quired the longitude. 

T. by W. 9 30 15 © ' 5 dec - E( l- °f *■ 

J o / // // m s s 

C—W +5 12 26 +19 23 40 +3^5 —3 51.1 +0.067 

Chro. cor. +25 15 +1 45 j 10O5 +.2 ( ^20 

G. m. t. May 17 3 7 56 = 3 d .13 + 19 25 25 I 4.4 3 50-9 ( 1 

— Eq. of t. +3 51 

G. ap. t. May 17 3 11 47 



L. ap. t. May 16 21 45 46 
Long. +5 26 1 1 



/ // 
O 58 17 


\ 


In. cor.+ 


/ // / // 
3 20 dip. —4 3 


+ 14 36 


( 


S.diam. + 15 50 ref.&p.-31 


h= 58 31 36 








L = 24 50 




1. sec 


0.04214 


p — 70 34 35 




1. cosec 


0.02545 


2s= 153 56 11 








s= 76 58 6 




1. cos 


9.35315 


— h = 18 26 30 




1. sin 


9.50015 
18.92089 


81° 30' W. 




1. sin -J 


t. 9.46045 



May 17, noon, lat. by mer. alt. of O, 25° 8' N". ; run of 
the ship from 9^ A. M., E. N". E. (true) 18 miles. 

For E. N. E. 18', I = 6'.9 N"., p = 16'.6 E., D = 18'.4 E. 

At the time of the A. M. observations, then, the latitude 
carried back from noon was 25° 1' N". Using this in the 
computation of the time, we find the L. ap. t. May 16, 
2ih 45m 29 s , and the long 81° 2 9 J' W. Applying D = 18'.4 
E., we have for the longitude, 

May 17, noon, 81° 11' W., from observations at 9.45 A. JI. 



A 



196 



NAVIGATION. 



By P. M. observations, and reduced to noon, the longitude 
was found to be, 

May 17, noon, 80° 44' W. from observations at 3.45 P. M. 
As the position is in the Gulf Stream, where there is a 
strong easterly current, the difference of the two longitudes 
is attributed to that cause. We take, then, as the longi- 
tude at noon, 



81° 11'- 



2.2x27' 



= 81° l'-W. 



2. At sea, 1865, Sept. 5, 4£ A. M., lat. 20° 16' S., long. 
74° 20' W. T. by Chro. 10 h 36 ra 25^ ; Chro. cor. (G. m. t.) 
— l h 16 m 10 s ; obs'd alt. JL, 20° 16' 0", W. of meridian; 
Index, cor. + 2' 20"; height of eye 15 feet; required the 
longitude. 



h 

T. by Chro. 12 
Chro. cor. 


h ra a 

+ 10 36 25 
— 1 16 10 


D'si2.A 

h m s 

22 40 41.9 


+ 2.370 


]) 's dec. 

o i n // 

—4 39 5 + 11.62 


G-. m. t. Sept. 


4 21 20 15 

+ 10 54 28.3 


+ 48.0 < 
22 41 29.9*1 


|47.4 
t - 6 


+ 355J 232 
—4 35 10 ( 3 


Red. for G. m. t 


+ 3 30.3 


o / // 

J. 20 16 | 


[ In. cor. 


+ 2 20 H. par. 60 39 


G. sid. t. 
$>'si2.JL 


8 18 13.6 
22 41 29.9 


+ 15 14" 
h'= 20 31 4 ' 


Dip —3 49 
[S.diam. + 16 33 +10" 


D 's G. h. ang. 


+ 9 36 44 


+ 54 19 


Par. and ref. 






h— 20 25 23 










L— 20 16 


1. sec 


0.02776 






p= 85 24 50 


1. cosec 


0.00139 






2 5=126 6 13 










8= 63 3 7 


1. cos 


9.65627 




s 


-h— 42 37 44 


1. sin 


9.83074 
19.51616 


D's L. h. ang. 


+4 39 37 




1. sin \ 


9.75808 


Long. 


+-4 57 7 


or 74° 17' W. 







Note. — The examples under Problem 30 can be adapted to this by regard- 
ing the chronometer correction given, instead of the longitude. 

200. 3d Method (Littrow's. By double altitudes of the 
sarne body.) 



LONGITUDE. 197 

When two altitudes of a body have been observed, and 
the times noted by the chronometer or watch, the hour- 
angles and local times can be found from each separately ; 
and thence the longitude for each. But we may also com- 
bine them and find the hour-angle for the middle instant be- 
tween them. 

Problem 52. From two altitudes of a heavenly body, sup- 
posing the declination to be the same for both, to find the 
mean of the tioo hour-angles, the latitude of the place and 
the Greenwich time being given. 

Solution. Take the mean of the two noted times and re- 
duce it to Greenwich mean time ; and find for it the declina- 
tion of the body. 

Reduce the observed altitudes to true altitudes. 

Let h and h' be the two altitudes, 

T&nd T r , the corresponding hour-angles ; 

then we have, by (116), 

sin h = sin L sin d + cos L cos d cos T, 
sin h'= sin L sin d + cos L cos d cos T' / 

and by subtracting the first from the second, 

sin h ! — sin h = cos L cos d (cos T'— cos T). 
By PL Trig. (106) and (108), this reduces to 

sin i (h'— h) cos i (h' + h)=—cos L cos d sin J (T + T) sin -J (T'—T) ; 
whence 

sin i (T'+ T) = - S ^- %7 h) T < : 0Si f' +h ] - 
* x ' sin \ (T — T) cos L cos d 

Put H Q = \ (A' + A), the mean of the two altitudes, 
T = i (T+T), the middle hour-angle, 
t = (T'—T), the difference of the two hour-angles ; 

and we have 

. m sin \ (h—h r ) „ _ - 

sin T == . \ + — cos H sec L sec d. (146) 

sm \ t u v ' 



198 NAVIGATION. 

t, for the sun is the elapsed apparent time ; for a star, the 
elapsed sidereal time ; and for the moon or a planet, the 
elapsed sidereal time — the increase of right ascension in 
the interval ; and can be found from the difference of the 
two chronometer times. 

Then, by (137), T can be found, and, as any other local 
hour-angle, subtracted from the corresponding Greenwich 
hour-angle, which in this case is to be derived from the 
mean of the noted times. 

T is + or — according as the second altitude is less or 
greater than the first ; so that it is on the same side of the 
meridian as the body at the time of its less altitude. , 

If (h — h!) is very small (146) becomes approximately 

sin To = Ha-aOrini 'coag. 

sin iK feoa I j ros a .■* v ' 

(148) 



sin \ t cos L cos d 
If T is very small, 

m _ sin \ Qi — li' ) cos H 



15 sin 1" sin J t cos L cos d y 
and if both are small, 

T _ i(h — h')cosH 

15 sin i t cos L cos d * 



(149) 



201. To estimate the effect of small errors in the data of 
the problem, let 

A (Ji—h') be a small increment of the difference of the two 

altitudes, 
A H Q , of the mean of the altitudes, 
A L, of the assumed latitude, 
A t, of the elapsed time, and 
A T„ the corresponding change of the middle hour-angle ; 

the last two expressed in seconds of time, the rest 

in minutes of arc : 



LONGITUDE. 199 

then, we have the formula,* 

Z, ^°-Ltani(A-^) 5 2tani*' cot ^ + cot ZJ tan V' li °° j 

in which each term may be computed separately for any 
supposed value of its numerator. The possible error in any 
case, on the suppositions made, would be the numerical sum 
of the several terms. 

As h—h\ however, is the change of altitude in the inter- 
val £, we may attribute all the error to h—h\ and regard t 
as exact ; or we may attribute all the error to t and regard 
h—hl as exact: so that one only of the first two terms in 
the second member of (150) is needed. 

When h—ti is so small that we may put cos-J- (h—h!) =1, 
we may use instead of (150) 

" T. = .\ J t {h - A ' ):C T ff \ , (151) 

sin -J- £ cos L cos a cos TJ v ' 

which is preferable to (150) and requires for computation 
four of the same logarithms as (147). 

The effects of errors are evidently least when T == ; 
that is, in the case of equal altitudes each side of the meri- 
dian. They increase rapidly as T increases; so that the 
method is especially adapted for altitudes nearly equal on 
both sides of the meridian, or for circum-meridian altitudes. 

But in the latter case, especially, a high altitude is neces- 
sary; for from (151) it appears that the effect of error in 

* Changing (146) into a logarithmic form, we have, 
1. sin T = 1. sin -J- (A — h') — 1. sin -J- t + 1. cos H Q — 1. cos L — 1. cos d ; 
differentiating each term except the last, and reducing, we obtain, 
d T Q d(A — /*/) dt _dH Q dZ 



tan T 2 tan i (h — h') 2 tan \ t cot B Q cot L' 

This is reduced to (150) by multiplying A (h — h' ), A H^ and A L by 4, 
to reduce minutes of arc to seconds of time. 



200 NAVIGATION. 

the difference of altitudes, h—h\ is least either when S is 
very near 90°, or t near 12 h ; so that, if t is small, H Q should 
be quite large. 

202. The method presents no special advantages for ob- 
servations on shore, except in the case of two nearly equal 
altitudes of a fixed star on opposite sides of the meridian. 
In the case of the sun and planets, it is necessary to take 
the change of declination into consideration, to obtain pre- 
cise results. 

The special case for which the method provides is at sea, 
within the tropics, when the sun passes the meridian at a 
high altitude. In that case, when by reason of clouds ob- 
servations near noon only can be made, or it is desired to 
obtain the longitude as near noon as practicable, let a pair 
of altitudes, or several pairs, be measured and the times 
noted with all tire precision practicable. The altitudes 
should be reduced to true altitudes, and one of each pair 
for the run of the ship in the interval* by the method given 
in Prob. 58, and in Bowd., p. 183. From each pair the 
middle apparent time can be found by (146), and the mean 
of these times subtracted from the mean of the Greenwich 
apparent times for the longitude. 

If the altitude changes uniformly with the time, or nearly 
so, the mean of several altitudes observed in quick succes- 
sion can be taken for a single altitude, 

If the observations have been made with care, the errors 
of instrument, refraction, and dip will affect the two alti- 
tudes of each pair nearly alike ; and if the reduction for the 
run of the ship is carefully made, the difference of altitudes 
in comparison with the difference of times will be nearly 
exact. 

203. This method was proposed by M. Littrow, Director 

* This may be avoided, if the course of the ship is at right angles to the 
bearing of the sun. 



LONGITUDE. 



201 



of the Vienna Observatory, and has been successfully used 
by Admiral Wullerstorf, of the Austrian navy, in 1857 and 
1 858. It is highly commended by M. Faye in a full discus- 
sion of it in the Comptes Rendus of the French Academy, 
March 7, 1864. It should be used cautiously, and the errors 
to which the result is liable in any case carefully estimated. 
Altitudes greater than 80° and an interval of more than 
half an hour are recommended, but an intelligent navigator 
can readily determine by (150), when he can safely depart 
from these limits. This will be especially the case when 
the altitudes are on both sides of the meridian. 



Examples. 

1. 1865, May 16, 11|- A. M., in lat. 25° 15' ST., long. 
56° 20' W., by account; the ship running K". E. (true) 8 
knots an hour. 

T. by Chro., 2 h 32 m 23 s Q's true alt., 81° 1' 0", 

" " " 2 53 11 " " " 83 40 30; 

Chronometer correction on G. mean time + 40 m 51 s ; re- 
quired the longitude. 

The distance sailed in the interval is 2'. 8. The sun's azimuth at the 1st 
observation is found to be N. 131° E., which differs 86° from the course. 
The reduction of the 1st altitude to the place of the 2d is (Prob. 58) 

2'.8 x cos 86° — + 0'.2 = + 12". 



J st chro. t. 




2 32 23 


O's dec. 


Eq. of t 


2d " " 
Elapsed chro. t. 
Mid. " " 
Chro. cor. 




2 53 11 

t - 20 48 

2 42 47 

+ 40 51 


O / " II 

+ 19 10 4 +34.3 

+ 1 57 ( 103. 
+ 19 12 1 1 14. 

o / // 


m s s 

—3 52.41 + 0.05 

+.17 
-3 52.2 


G. m. t. May 16 




3 23 38 


h — 81 1 12 




— Eq. of t. 




+ 3 52 


h' — 83 40 30 




G. ap. t. 




3 27 30 i (A— h')=- r 1 19 39 


1. sin 8.3649 n 


L. ap. t. 




23 41 44 


H Q = 82 21 


1. cos 9.1242 


Long, at 2d obs. 


I 


+ 3 45 46 

56°26'.2W. 


L= 25 15 
d— 19 12 


1. sec 0.0436 
1. sec 0.0249 


Red. for h .l 




0.6 E. 


t= h 20 m 48 s 


l.cosec-J-2 1.3433 


Long, at noon 




56 27 W. 


T Q =-0 18 16 


1. sin 8.9009 n 



202 



NAVIGATION. 



By (150), if A (h-N)bz+l\ A 7; = +6».9; if AB = + 1\ 
A 7 7 = +2s.4; if JZ = + 10', A T Q = + 1K5. 

2. At sea, 1865, June 29, lat. at noon by mer. alt. of O, 
33° 25' N., long, by account 147° 10' E. ; 
near 11 A.M. T. by Chro., 1*55™ 54° ) , fltruealt . M o 2r 30 , ; 

" 1 P. M. " " " 3 45 ) 
Chro. cor. on G. m. t. — 36 m 28 s ; the ship run 

from 1st observation, to noon, N". 3 pts. W. 9'.9 ) 
" noon to 2d observation, N. 2 " W. 7 .2 ; ) 
required the longitude at noon. 



N. 3 W. 

N. 2 W. 


9'.9 

7.2, 


8'.2 N. 
6.6 


5'.5 W. 

2.8 


JA=6'.6W. 
3.4 


N. 30° W. 


17.0 


14.8 


8.3 


10.0 



The sun's azimuth was found to be N. 127° E. at the 1st observation ; 
N. 127° W. at the 2d observation. 

The difference of N. 30° W. and N. 127° E. is 157° ; 
the difference of S. 30° E. and N. 127° W, is 83°. 
It will be better, therefore, to reduce the second altitude to the position 
of the first. By Prob. 58, (or Bowd, p. 183,) this reduction is I7'.0xcos 



83° = + 2'.1. 


The latitude at 


the time of the 1st observation was 


33° 16'.8 N. 


h m s 




A. M. chro. t. 


13 55 54 


O's dec. Eq. of t. 


P. M. M " 


15 45 


+23° 16' 59"—- 7".3 +2 m 55 9 .6+0 8 .507 


Elapsed " " 


t— 1 49 6 


— 1 45 +7.2 ( 7 .10 
+ 23 15 14 +3 2 .8 ( .11 


Mid. " " 


14 50 27 
— 36 28 


Chro. cor. 


h = 74° 21' 30" 1. cosec j t 0.627 


Mid.G.m.t. June 28 14 13 59 


h' — 74 23 36 log T l T 8.824 


— Eq. of t. 


__ 3 3 %(h—h')= —1 3 log 1.799 n 


Mid. G. ap. t. 


14 10 56 


H Q - 74 22 33 1. cos 9.430 


" L. ap. t. 


23 59 54 


L = 33 17 1. sec 0.078 


Long, at 1st obs 


( —9 48 58 
'( 147° 14'.5 E. 


d = 23 15 1. sec 0.037 
T = — 6 9 .3 log 0.795* 


Red. to noon 


6 .6 W, 




Long, at noon 


147 7 .9 E. 





LONGITUDE. 203 

By (151) if A (h—h ! ) = 1', A T = 3 s .l. It would require 
a change of l£° in i, or of 2-|° in jET , for either to change 
T one second of time. The accuracy of the result, there- 
fore, depends upon the accuracy with which the difference 
of altitudes has been found ; that is, in this case, mainly 
upon the course and distance made good. 

204. 4th method. (By equal altitudes.) Let equal altitudes 
of a heavenly body be observed east and west of the meri- 
dian (Art. 175) and the times noted as in other observa- 
tions ; and the mean of the watch-times in each set, if a 
watch is used, reduced to chronometer time. If both sets 
have been observed at the same place, and the declination 
of the body has not changed, the mean of the two times 
will be the chronometer time of its meridian transit. 

If the declination has changed in the interval, as is ordi- 
narily the case with the sun, moon, or a planet, the correc- 
tion for such change, found by the methods of Problem 51, 
should be applied. 

Applying then the chronometer correction, we have the 
corresponding Greenwich time, which will be mean or side- 
real as the time to which the chronometer is regulated. 

Finding from this, by the method in Art. 189, the Green- 
wich hour-angle of the body (which in the case of the sun 
is the Greenwich apparent time), we have the longitude, if 
the first observation was east of the meridian, as the cor- 
responding local hour-angle is then 0. But if the first ob- 
servation was west of the meridian, the local hour-angle is 
12 h and must be subtracted. 

This method should be used on shore, when practicable, 
in preference to either of the preceding. 

205. Equal altitudes of the sun can be conveniently used 
at sea w T hen the sun passes the meridian near the zenith ; 
that is, when its declination and the latitude are nearly the 
same. Altitudes very near noon are then available for find- 
ing the time (Art. 196), and equal altitudes can be observed 



201 NAViGvrio^. 

with only a short interval. In the example of Art. 196, an 
interval of eight minutes would have been sufficient. 

If the ship does not change her position in the interval, 
the middle time corresponds to apparent noon ; as the change 
of declination may be neglected, unless the interval between 
the observations is so great as to require it. 

206. If the longitude only has changed, the middle time 
corresponds to apparent noon at the middle meridian, and 
will give the longitude of that meridian. This will be the 
longitude at noon, if the speed of the ship has been uniform. 
But if it has not. subtracting half the change of longitude, 
when the true course is icest, or adding it when the course 
is east, will give the longitude of the place where the first 
altitude w T as observed. This can then be reduced to noon 
by allowing for the run of the ship. 

If the change of longitude is west, the sun arrives at the 
corresponding altitude of the afternoon later than it would 
do if observed at the same place as in the forenoon ; if the 
change is east, it arrives earlier ; and the difference is the 
time of the sun's passing from the one meridian to the other, 
that is, the difference of longitude expressed in time. 

If, then, 2 t is the elapsed apparent time, 

A A, the change of longitude (+ when west), the 
hour-angle of the sun at each observation is t — \ A X ; and 
(137) becomes 

a rp_ A h d. t tan L A h d. t tan d 

°~~ 15 sin (t—i JA) + 15 tan (t~i A X)' ^ Z ' 

But even when the elapsed time is so great that it is thought 
necessary to correct for the change of declination, A X is never 
large enough to produce a change of I s . 

If the latitude only has changed, the middle time requires 
correction for such a change, which can be deduced in a 
similar way to that for a change of declination in Prob. 51. 
But, as in the fundamental formula, 



LONGITUDE. 205 

sin h = sin JL sin d + cos L cos d cos t, 

L and c? enter with the same functions, they are interchange- 
able. If, then, 

J h X is the hourly change of latitude (+ toward the north 

and expressed in seconds), and 
A ' T^ the required correction, 
we have from (137) and (139), 

d'T =- A f- tUm f d + *£?*;* (153) 

lo sin t lo tan t v ' 

and A'T Q —A A h Z. tan d+B J h Z. tan X, (154) 

for which Chauvenet's tables can be used. 

If both latitude and longitude have changed, for t in the 
denominators of (153), we may substitute t — \ AX\ but 
this at sea is a needless refinement. 

The restriction of this method to a short interval between 
the observations, depends upon the uncertainty of the run of 
the ship and consequent imperfect determination of ^ h i, the 
mean hourly change of latitude in the interval. If its error 
is supposed to be - A h L, the consequent error in A' T is - A r T Q . 

When equal altitudes near noon are practicable, a merid- 
ian altitude of the sun can ordinarily be taken for latitude, 
so that L will be sufficiently exact. Moreover, the latitude 
and longitude are both found for noon. 

Examples. 

1. At sea, 1865, March 17, noon, rat. by mer. alt. of the 
sun 3° 16' S., long, by account 84° 58' W. ; equal altitudes 
of the sun were observed at 5 h 34 m 18 s and 6 h 3 m 24 s 
G. mean time ; the ship running S. S. E. (true) 10 knots an 
hour ; required the longitude. 

For S. S. E., 10', J h i = - 9'.2, J h X = — 3'.8 



206 NAVIGATION. 



h 



1st G. m. t. March 17 5 34 18 0'* dec. Eq'n oft. 

2dG. m. t. 6 3 24 — 1° 13' 10" + 59".25 +8 m 27 9 .5-0 9 .736 

Elapsed time . 29 6 + 5 44 j 296. — 4 .3 ( 3 .7 

Mid. G. m. t. March 17 5 48 51 - 1 7 26 I 48. + 8 23.2 ( -6 

— Eq. oft. —8 23 j h i = - 552" log 2.742 w log 2.742n 

Mid. G. ap. t. March 17 5 40 28 L=— 3° 16' l.tan 8.756 n 

Red. for A L +5 rf = -l Utan 8.290 n 

j G. ap. t. of noon 5 40 33 • log A 9.406 n log£9.405 



or long. 85° 8'W. { — 2 S .7 log 0.438 n 

log 0.903 



( — 2 S .7 log CM 
( +8.0 



In this example the sun's azimuth was 120°, and in l m the 
altitude changed 13'. An inequality of 30" in the altitudes 
would therefore affect the result only T \ of l m , or l s .2. An 
error of 1' in the hourly change of latitude would affect the 

result — , or S .6. 

2. At sea, 1865, June 16, lat. at noon by mer. alt. of O, 
22° 50' 1ST., long, by account 35° 59' W. ; equal altitudes of the 
sun were observed at 2 h 16 m 18 s and 2 h 31 m 42 s G. mean time ; 
the ship running S. (true) 14' an hour. 

O's dec. +23° 22'.4 



Elapsed time 


h 15 m 24 8 


Mid. G. m. t. June 16 


2 24 


— Eq. of t. 


—22.5 


Mid. G. ap. t. 


2 23 37.5 


Red. for A L 


+ 2.5 


G. ap. time of noon 


2 23 40 


Long. 


35° 45' W. 



The sun's azimuth was 72°; the change of altitude in l m was 

13 '.2, so that an inequality of l' in the altitudes would affect 

the result ^\ of l m , or 2 S .3. An error of 1' in A h L would 

2 s 5 
affect the result —, or s . 2. 

14: ' 

3. At sea, 1865, June 29, h ; lat. by mer. alt. of 0, 33° 25' N., 
long, by account 147° 10' E. ; 



LONGITUDE. 207 

Chro. cor. on G. m. t. — 36 m 28 s ; In. cor. of sex't +0' 50"; 
height of eye 18 feet. The ship run 
from 11 A. M. to noon N. 3 p'ts W. 11' 
from noon to 3 P. M. N. 2 " W. 8' 
required the longitude at noon. 



For N. 3 W. 


11' jZ = +9'.l JX 


= + 7'.4 


N. 2 W. 


8 A£ = +1A zU 


= +3.7 


whence A L = + 8 .25 = 495" 




A. M. chro. t. + 12 h 


h m a 

13 55 54 Q'sdec. 


Eq'n of t. 


P. M. chro. t. 


15 45 +23° 16' 59"-7".3 


+ 2 m 55 s .6+0 9 .51 


Elapsed time 


1 49 6 —1 45 


+ 7 .2 ( 7.14 
+ 2 2.8 ( .11 


Mid. chro. t. 


14 50 27 +23 15 14 


Chro. cor. (G. m. t.) 


— 36 28 




Mid. G. m. t. June 28 


14 13 59 log J h L 2.695 


log J h L 2.695 


— Eq. of t. 


— 3 3 log A 9.410^1 


log B 9.398 


Mid. G. ap. t. 


14 10 56 1. tan d 9.633 


1. tan L 9.819 


Red. for A L 


+ 27 log 1.738 n 


log 1.912 


G. ap. t. of noon 


14 11 23 -54 s .7+81 s . 


7= + 27 3 


Middle long. \ 

( or 


-9 48 37 
147° 9'.2 E. 




Red. to noon 


1.8 W. 




Long, at noon 


147 7 .4 E. 





The sun's azimuth was 127° ; for A t = l m , A h = 10", and 
an inequality of 1' in the altitudes will effect the result ^ 
of l ra , or 3 s . An error of 1' in A h Z will affect the result 

27* 

These observations reduced as single altitudes, give, as 
the longitude at noon, 147° 7'.8 E. ; reduced by Littrow's 
method (Ex. 2, p. 202), 147° 7'.9 E. 

207. 5th Method (By transits.) 

Observe the transits of the sun or a star across the threads 
of a well-adjusted transit instrument, noting the times. Re- 



208 NAVIGATION. 

duce the mean of the noted times for semi-diameter and 
errors of the instrument as in Art. 184 ; and thence find the 
Greenwich hour-angle of the body in the way described in 
Art. 189. This will be the longitude, if the upper culmina- 
tion has been observed, as the local hour-angle is 0. If the 
lower culmination has been observed, the local hour-angle is 

This method can be used only on shore. 

Example. 

1865, May 17, I7 h 16 m 20 8 .5 G. mean time, the meridian 
transit of a Bootis (Arcturus) was observed ; required the 
longitude of the place of observation. 



G. mean time May 17 


11 h 16 ra 


20 8 .5 


£» 


3 40 


47.20 


Red. for G. m. t, 


+ 2 


50.24 


G. sid. t. 


20 59 


57.94 


*'sR.A. 


14 9 


32.81 


* 's H. angle or Long. 


+ 6 50 


25.1 or 102° 36' 17" W. 



LONGITUDE. LUNAR DISTANCES. 

208. Problem 53. To find the longitude by the distance 
of the moon from some other celestial object. 

Solutio7i. If we have given the local mean time and the 
true distance of the moon from some celestial object as seen 
from the centre of the earth, we may find, by interpolating 
the Nautical Almanac lunar distances (Prob. 28), the Green- 
wich mean time corresponding to this distance. The differ- 
ence of this from the local time is the longitude. 

The local time may be found for the instant of observation, 
either from an altitude of a celestial object observed at the 
same time, or by a chronometer regulated to the local time. 

At sea the correction of the chronometer on local time can 



LONGITUDE. — LUNAR DISTANCES. 209 

be found from altitudes observed near the time of measuring 
the lunar distance, and reduced for the change of longitude 
in the interval by the formula (Art. 167), 

c r — c + AX, 

A X being in time and + when the change is west. 

In practice, the apparent distance of the moon's bright 
limb from the sun or a star is observed, and the true distance 
derived by calculation, as in the next problem. 

209. Problem 54. Given the apparent distance of the 
moorts bright limb from a star, the centre of a planet, or 
the surfs nearest limb, to find the true distance of the moorfs 
centre from the star, or the centre of the planet or the sun. 

Solution. It is necessary that the altitudes of the two 
bodies should be known, either directly from observations 
at the same time, or from observations before and after, and 
interpolated to the time of observation (Bowd., p. 246) ; or 
computed from the local time (Prob. 38), (Bowd., pp. 247, 
&c). 

The Greenwich time is also supposed to be known ap- 
proximately, either from the local time and approximate 
longitude, or, as is preferable, from the time noted by a 
Greenwich chronometer. 

A complete record of the observations will include the ap- 
proximate latitude and longitude of the place, the local time 
and chronometer correction, the index corrections of the in- 
struments used, the height of the barometer and thermome- 
ter, and at sea, the height of the eye above the water, as 
well as the noted times of observation and the observed dis- 
tances and altitudes. Several observations may be made at 
brief intervals, and the means taken. 

210. The preparation of the data embraces : 

1. Finding the Greenwich mean time approximately from 
the chronometer time, or from the local time. 

2. Taking from the Almanac for this time the semi-diame- 



210 NAVIGATION. 

ter and horizontal parallax of the moon, and of the other 
body* when they are of sensible magnitude ; adding to the 
moon's semi-diameter its augmentation. (Art. 60.) 

At low altitudes the contractions produced by refractions 
should be subtracted from the semi-diameters of the sun and 
moon. Formulas for finding these are given in Art. 213. 

When the spheroidal form of the earth is taken into con- 
sideration, to the moon's equatorial horizontal parallax (Art. 
57), as taken from the Almanac, should be added the aug- 
mentation to reduce to the latitude of the place, which is 
found in Tab. III. of Chauvenet's Method. The declinations 
of the two bodies to the nearest decree are reauired from 
the Almanac for this purpose. 

3. Applying to the observed distance the index correction 
of the instrument, and, when the sun is used, adding the 
moon's augmented semi-diameter and the sun's semi-diame- 
ter ; when a planet or star is used, adding the moon's aug- 
mented semi-diameter if its nearest limb is observed, but 
subtracting it if the farthest limb is observed. 

4. Applying to the observed altitude of each body the 
index correction, dip, and semi-diameter (when necessary), 
so as to find the apparent altitude of its centre. If the true 
altitude is computed, the parallax must be subtracted and 
the refraction added. 

In the following direct method it is necessary also to find 
the true altitudes. 

211. To find the true distance, 
let D = the apparent distance of the centres, 

D'= the approximate true distance, 

h — the apparent altitude 1 n ^ , 

_, , A1 1 . _, y of S's centre, 

li = the true altitude ) 

_H= the apparent altitude ) r ^, , 

1 ^ . ^ I of s centre, planet, or star. 

II— the true altitude J \ 

* The sun's horizontal parallax may be taken as 8".5. 



LONGITUDE.— LUNAR DISTANCES. 



211 



In Fig. 35, let m and S be the apparent places of the 
moon and other body ; m 1 and S', their true places. 

The true and apparent places of each are on the same ver- 
tical circle, Z m, Z S respectively, since they differ only by 
refraction and parallax, which act only in vertical circles, 
except so far as a small term of the moon's parallax is con- 
cerned, which will be subsequently considered. 

Z Then m S = -Z>, the apparent 

distance ; 

m r S'=jD', the true distance ; 




and in the triangle m Z S, 
h Uei 



Fig. 35. 



m S = JD 

Z m = 90°— h Y being given, 
ZS = 90°--Zn 
to find the angle Z, we have by Sph. Trig. (32),* 
2 cos j Qi + #+ D) cos j (ft + E-L) 

4 cos h cos i/ 

Then in the triangle m' Z S', 

Z m'= 90°- A' and Z S'= 90°-J7 ; 
being given, m! S' may be found by Sph. Trig. (I7),f 

sin 2 i D f = cos 2 1 {h' + H f ) - cos A' cos .fl 7 cos 2 | Z, 
or by substituting the value of cos 2 £ Z, and putting 

5 = 1 (A + JST+Z>), (155) 

sin 8 \ D'=cos 2 \ (h'+ H') - cog h cosH cos s cos (s-D). 

To adapt this for logarithmic computation put 

. 21 cos h' cos H' . _. , ^ . 

sin f m = — — 7 ^ cos 5 cos (s— x>), (156) 

^ cos A cos R v y ' v J 



cos 2 -J ^L 



_ sin -J- (a+6+c) sin J (6+c— a) 



sin 6 sin c 
f sin 2 i a = sin 2 •£■ (6-f-c) — sin 6 sin <? cos 2 -J ^4. 



212 
then 



NAVIGATION". 



sin* 2 i &'= cos 2 \ (h' + H f ) - sin 2 % m, 

which by PL Trig. (134), becomes 

sin 2 I D r = cos i (h' + H'+ m) cos J (A'+Jff 7 - m), 

or, if we put 

s , =±i(h f +M , + m) J (151) 



we have 



sin \ D f = |/[cos $' cos (5'— m)]. 



(158) 



The solution is effected by formulas (155), (156), (157), 
and (158). 

This is only one of several direct trigonometric solutions. 
It is easily remembered, involving only cosines in the second 
members. But in all such methods 7-place logarithms are 
required for the computations. 

212. If the moon's augmented parallax has been used, the 
distance obtained, D\ is not the true 
distance as seen from the centre of 
the earth, but from the point C (Fig. 
36), where the vertical line of the 
place intersects the earth's axis. 

A reduction to the centre, (7, is 
still required, for wdiich we have the 
formula — * 




AD'=Att sin L 



sm 



sin 6„ 



(159) 



Vsin D' tanDV' 
in which 

S 8 is the sun's declination, 
d ro , the moon's declination, 
7T, the moon's equatorial horizontal parallax, whose mean 

value is 51' 30", 
A, a coefficient depending on the eccentricity of the terres- 



* Chauvenet's Astronomy, Vol. I., p. 399. 



LONGITUDE. — LUNAR DISTANCES. 213 

trial meridian, the mean value of which, for latitude 45°, 

is .0066855, or of log A, 7.8251, 
A sin i, the distance C C, with CE = 1. 

The mean values of An = 23".07, or log A tt = 1.3630, 
may be used, unless great precision is required. 

The signs of the declinations and latitude are + when 
north, and A D' is to be added algebraically to D'. 

If the augmentation of the parallax has been neglected, 
the distance has been reduced to a point on the vertical line 
between C and C" and at a distance from A equal to the 
equatorial radius C E. 

213. To find the corrections needed for the contraction by 
refraction of the semi-diameters of the sun and moon in the 
direction in which the distance is measured, 

let q ■=■ the angle Z S m (Fig. 35), at the sun or star, 
Q z=z the angle Z m S, at the moon, 
A s and A's, the contractions of the sun's semi-diameter 

respectively in the vertical direction S Z, and in the 

direction of the distance S m / 
A S and A'S, the contractions of the moon's semi-diameter 

respectively in the vertical direction m Z, and in the 

direction of the distance m S. 

To find q and Q from the three sides of the triangle Z S m, 
putting, as in (155), 



we have 



. - //cos s sin (s—H)\ 

S1D ^ = fi sin 2? cos A j 



sin D cos H 



(160) 



for which it will suffice to use a rough approximation of Z>, 
and for the computation logarithms to four places ; as q and Q 
are required only within 30'. 

The contractions, A s and A jS, of the vertical semi-diame- 



214 NAVIGATION. 

ters may eacli be found from the refraction table, by taking 
the difference of refractions for the limb and centre. 

Then, for the required corrections, we have the formulas,* 

A's = As cos 2 q, A f $ = AjS cos* Q. (161) 

This contraction for either body is less than l", if the alti- 
tude is greater than 40°. For a very low altitude, it is best 
to subtract it from the semi-diameter m the preparation of 
the data, so that D will be corrected for it. But, unless 
quite large, it will suffice to compute it subsequently, and 
subtract it from D' when the nearest limb is used, or add it 
to D' when the farthest limb is used. 

214. Let A D = the reduction of the apparent distance to 
the true, or I) '= D + A B. 

A great variety of methods have been given for finding 
A Z>, requiring 4 or, at the most, 5-place logarithms ; but 
also needing special tables. Four such methods are con- 
tained in Bowditch's Navigator. They generally neglect to 
take into account the spheroidal form of the earth, the cor- 
rection of refraction for the barometer and thermometer, 
and the contraction of the semi-diameters of the sun and 
moon. 

These together, at very low altitudes and in extreme cases, 
may produce an error of 3 m in the calculated Greenwich 
time, and do actually, in the average of cases, produce errors 
from 10 s to l m . 

Prof. Chauvenet has given in the American Ephemeris for 
1855,f a new form to the problem, with convenient tables, 
by which all these corrections are readily introduced. It is 
but little longer than the other approximative methods, in 
which they are neglected. 



* Chauvenet's Astronomy, Yol. I., p. 186. 

\ Reprinted in a pamphlet with his method of equal altitudes. 



LONGITUDE. — LUNAR DISTANCES. 215 

215. The moon's mean change of longitude is 13°. 17640 
in a clay (HerschePs Ast., p. 222), or 33" in l m of time. 

An error, then, of 33" in the distance will, in the average, 
produce an error of l m in the Greenwich time, or 15' in the 
longitude ; or an error of 10" in the distance will produce 
an error of about 20 s in the Greenwich time, or 5' in the 
longitude. 

We may, however, readily find the effect of an error of 
1", and thence any number of seconds, in the distance, by 
taking the number corresponding in a table of common log- 
arithms to the " Prop. Log. of Diff." in the Almanac ; for 
this prop. log. is simply the logarithm of the change of time 
in seconds for a change of l" in the distance, (p. 95.) 

216. Errors of observation are diminished by making a 
number of measurements of the distance. But even with a 
skilful observer a single set of distances is liable to a possi- 
ble error of 10" or even 20". 

Errors of the instrument are diminished by combining re- 
sults from distances of different magnitudes, especially from 
those measured on opposite sides of the moon. This cannot 
usually be done with longitudes at sea, but may be with de- 
terminations of the chronometer correction. The error pe- 
culiar to the observer, that is, in making the contacts always 
too close, or always too open, is not eliminated in this way, 
but will remain as a constant error of his results. 

The accuracy of the reductions of the observed to the true 
distance, depends more upon the precision w^ith which the 
differences of the apparent and true altitudes — that is, the 
parallax and refraction — have been introduced, than upon 
the accuracy of the altitudes themselves. 

217. Lunar distances are used at the present day, not so 
much for finding the longitude, as for finding the Greenwich 
mean time, with which to compare the chronometer. They 
may thus serve as checks upon it, which in protracted 



216 NAVIGATION. 

voyages may be much needed. If the chronometer cor- 
rection thus determined agrees with that derived from the 
original correction and rate, the chronometer has run well, 
and its rate is confirmed ; if otherwise, more or less doubt is 
thrown upon the chronometer, according to the degree of 
accuracy of the lunar observation itself. If the discordance 
is not more than 20 s , it is well still to trust the chronometer, 
as the best observed single set of distances may give a result 
in error to that extent. If it is large, then by repeated 
measurements of lunar distances, differing in magnitude, 
and especially on both sides of the moon, and carefully re- 
duced, the chronometer correction can be found quite satis- 
factorily. By taking the rate into consideration, observa- 
tions running through a number of days can be combined. 

Example. 

At sea, 1855, Sept. 7, about 6 h A. M., in lat. 35° 30' 1ST., 
long. 30° W. by account ; 

Time by chro. 8 h 29 m 57 9 .5 ; app. chro. cor. (G.m.t.) — 21 m l s .5 ; 
Observed distance of © and S> 43° 52' 30", index cor. — 20"; 
Observed altitude of JD 49° 31 ' 50", index cor. + 1' 0" ; 

Observed altitude of O 5° 21' 10", index cor. 0"; 

Bar. 29.10 inches ; ther. 75° ; height of eye 20 feet ; 
Required from these observations the chronometer correction 
on Greenwich time. 

Preparation. 

h m s / // // 

T.bychro. 12 h +8 29 57.5 D'sH.par. 54 19.4 D'sS.diani. 14 50.0 

Chro. cor. -21 1.5 Aug. +3.6 Aug. +11.2 

G. m. t. Sept. 6 20 8 56 J) >s Aug. H. par. 54 23.0 J) 's Aug. S. diam. 15 1.2 

O's H. par. 8 ".5 

0's S. diam. 15' 55". 1 
V. cont. -21 .6 

V. S. diam. 15 33 .5 



_£ 


49° 31' 50" 


0. 5° 27' 10' 


Iu. cor. 


+ 1 o 


In. cor. 


Dip 


—4 23 


Dip -4 23 


Aug. S. diam. 


+ 15 1 


V. S. diam. +15 34 



LONGITUDE. — LUNAR DISTANCES. 217 

H = 49° 43" 28' h = 5° 38" 21' 

Ref. —46 Ref. -8 12 

Par. +35 10 Par. +8 

#'=50 17 52 h'= 5 30 17 

Obs'd dist, J) O 43° 52' 10' 
il= 49° 43' 3>'s Aug. S. diam. +15 1.2 

h= 5 38 1. sec 0.002 _ *, ° . « «i 

D = 44 22 1. cosec 0.155 s S ' dlam ^" 15 5oJ 

2 s = 99 43 Cont. A's = —21.6 

s = 49 52 1. cos 9.809 D = 44 22 45 

&-H= 9 1. sin 1418 Cont. Q's S. diam. (Ml). 

17.384 2 log cos a 9.999 

q = 2_J9 1. sin J 8.692 lo | 2 1.6 1.334 

log A's 1.333 



Computation of True Distance. (155-158) 

O I II O I II 

#=49 43 28 LsecjBT 0.1894554 

h = 5 38 21 1. sec h 0.0021069 

D = 44 22 45 #'= 50 17 52 1. cos H' 9.8053633 

2 s = 99 44 34 h' = 5 30 17 L cos h' 9.9979925 

s = 49 52 17 1. cos s 9.8092266 

s — D= 5 29 32 o i a 1. cos (s-D) 9.998001 7 

Compression. (159) 4(^+^=27 54 4.5 19.8021464 

O'sdec :=+ 6°.3 L sin 9.040 * W= 52 46 39 ' 4 "-* 1 *" 1 9 ' 9Q1Q732 

Z>' 1. cosec 0.150 s'= 80 40 44 

«== 0.155 log 9.190 m=105 33 19 L cos s' 9.2094277 

D'sdec.= +25°.3 1. sin 9.631 m— s'= 24 52 35 1. cos (m-s') 9.9577114 
D' 1. cot 9.999 o t u 19.1671391 

n'= .427 log ^630 j J)' = 22 32*23.? 1. sin 9.5835696 

n-n' = - .272 log 9.435* .p = ^ ^^ 

Aiz log 1.363 n 

L 1. sin 9.764 Cor. f or Com P- ~ 4 

Comp.— 3". 6 log 0.562 n D"— 45 4 43 true distance. 

Finding the Greenwich mean time and chronometer cor- 
rection. (82) 

True distance D"= 45° 4' 43" 

Distance at 18\ D = 46 3 17 P. L. 0.3433 Diff. +5 

J)"— D Q = 58 34 log 3.5458 

tz= 2 h 9 m 6 8 log 3.8891 



218 





NAVIGATION. 


G. m. t. of D 


18 h m s 


Red. for 2d diff. 


-2 


G. mean time, Sept. 


6 20 9 4 


T. by chro. 


20 29 57 


Chro. cor. 


— 20 53 by lunar. 




— 21 1 by previous cor. and rate. 


Difference 


+ 8 



This example is taken from the pamphlet of Prof. Chauvenet, where it is 
reduced by his method with far less labor of computation. The true distance 
by that method is 45° 4' 45" ; by Bowditch's 1st method, in which the small 
corrections are omitted, it is 45° 5' 44", differing very nearly 1' from the 
correct value. This would produce an error of 2 m 10 s in the Greenwich 
time. 

218. Other lunar methods for finding the longitude, be- 
side that of lunar distances, are — 

1. By moon culminations, or observing the meridian 
transits of the moon and several selected stars near its path, 
whose right ascensions are considered well determined. 

2. By occidtations, or noting the instant that a star dis- 
appears by being eclipsed by the moon, or that it reappears 
from behind the moon. The first is called an immersion, 
the second an emersion. 

3. By altitudes of the moon near the prime vertical. 

4. By azimuths of the moon and stars observed near the 
meridian. 

These methods, except occasionally the second, are avail- 
able only on shore. They require good instruments, careful 
observations and determinations of the instrument correc- 
tions, and scrupulous exactness in the reductions, especially 
those which involve the moon's parallax. 

By each may be found the moon's right ascension, and 
thence, by inverse interpolation in the Almanac, the corre- 
sponding Greenwich mean time. Subtracting from it the 
local mean time, which must also be found from good ob- 
servations, gives the longitude. 

219. If corresponding observations are made at two dif- 



LONGITUDE. — LUNARS. 219 

ferent places, their difference of longitude can be found with 
much less dependence on the accuracy of the Ephemeris. 

When the two local times of the occupation of the same 
star have been noted, they can each be reduced to the in- 
stant of the geocentric conjunction of the moon's centre and 
the star in right ascension ; and the difference of the reduced 
times will be the longitude. 

By the other methods, the change of the right ascension 
of the moon, in passing from one meridian to the other, may 
be found. This, divided by the mean change in a unit of 
time, as l h or l m , computed from the Ephemeris, will give 
the difference of longitude in the same unit. 



CHAPTER IX. 

SUMNER'S METHOD: LATITUDE AND LONGITUDE BY 
DOUBLE ALTITUDES. 

CIRCLES OF EQUAL ALTITUDE. (SUMNER'S METHOD.) 

219. Suppose that at a given in- 
stant the sun, or any other heavenly 
body, is in the zenith of the place M 
(Fig. 3 7) , on the earth ; arid let A A 7 A" 
be a small circle described from M as 
a pole. The zenith distance of the 
body will be the same at all places on 
this small circle, namely, the arc MA; 
for if the representation is transferred 
to the celestial sphere, or projected on the celestial sphere 
from the centre as the projecting point, 

M will be the place of the sun, or other body, and the circle 

A A' A" will pass through the zeniths of all places on the 

terrestrial circle, and 
M A, M A', &c, will be equal zenith distances. 

The altitude of the body will also be the same at all places 
on the terrestrial circle AAA"; hence such a circle is called 
a circle of equal altitude. 

It is evident that this circle will be smaller the greater 
the altitude of the body. 

220. The latitude of M is equal to the declination of the 
body, and its longitude is the Greenwich hour-angle of the 




Fig. 37. 



CIRCLES OF EQUAL ALTITUDES. 221 

body ; which, in the c&se of the sun, is the Greenwich appar- 
ent time, or 24 h — that apparent time, according as the time 
is less or greater than 12 h . This is evident from the dia- 
gram, in which, regarded as on the celestial sphere, 
P M is the celestial meridian of the place, whose zenith is 

M, and its co-latitude ; and also the declination circle, and 

co-declination, of the body M ; 
and if P G is the celestial meridian of Greenwich, 6PM is, 
at the same time, the longitude of the place, and the Green- 
wich hour-angle of the body. 

If, then, the Greenwich time is known, the position of M 
may be found and marked on an artificial globe. 

221. If, moreover, the altitude of the body is measured, 
and a small circle is described on the globe about M as a 
pole, with the complement of the altitude as the polar radius, 
the position of the observer will be at some point of this cir- 
cle. His position, then, is just as well determined as if he 
knew his latitude alone, or his longitude alone ; since a know- 
ledge of only one of these elements simply determines his 
position to be on a particular circle, without fixing upon any 
point of that circle. 

As, however, he may be presumed to know his latitude 
and longitude approximately, he will know that his position 
is within a limited portion of this circle. Such portion only 
he need consider. It is commonly called a line of posi- 
tion.* 

222. The direction of this line at any point is at right 
angles with the direction of the body ; for the polar radius 
M A is perpendicular to the circle A A' A" at A, A', A", and 
every other point of the circle. 

223. Artificial globes are constructed on so small a scale 
that the projection of a circle of equal altitude on a globe 

* Inappropriately termed a line of bearing. 



222 NAVIGATION. 

would give only a rough determination. But the projection 
of a limited portion maybe made upon a chart by finding as 
many points of the curve as may be necessary, and, having 
plotted them upon the chart, tracing the curve through them. 
The portion required is usually so limited that, when the 
altitude of the body is not very great, it may be regarded as 
a straight line ; and hence two points suffice. With high 
altitudes, three points, or if the body is very near the zenith, 
four may be necessary, and even the entire circle may be 
required. 

224. Problem 55. From an altitude of a heavenly body 
to find the line of position of the observer ', the Greenwich 
time of the observation being known. 

Solution. From the given altitude, and assumed latitudes 
X n X 2 , X 3 , &c, differing but little from the supposed lati- 
tude, find the corresponding local times (Prob. 43), and 
thence, by the Greenwich time, the longitudes A x , A 2 , ^35 <&c. 
Thus we shall have the several points, whose positions are 
conveniently designated as (X 19 a x ,), (Z 2 , a 2 ,), (£ 3 , A 3 ,), &c. 

It facilitates the computation to assume latitudes differing 
10' or 20', as the % sums and remainders differ 5' or 10', and 
only one of each need be written. 

Or, from the Greenwich time and assumed longitudes, 
^1? ^2> ^3? &c., find the corresponding local times (Art. 77), 
and thence the hour-angles of the body (Probs. 34, 35). With 
these and the observed altitude, find the corresponding lati- 
tudes , i n i 2 , i 3 , &c. (Prob. 46). 

This is more convenient than the preceding method, when 
the body is near the meridian. 

In either mode the computation may be arranged so that 
the like quantities in the several sets shall be in the same 
line, and taken out at the same opening of the tables. 

The several points may then be plotted on a chart, each 
by its latitude and longitude, and a line traced through 



CIRCLES OF EQUAL ALTITUDES. 223 

them, which, will be the required line of position. Two 
points connected by a straight line are sufficient, unless the 
altitude is very great, or the points widely distant. 

Thus in (Fig. 38), let A and B be two BjL_L 

such points plotted respectively on the 

parallels of latitude L 15 L 2 , and each in ,// 

its proper longitude ; A B is the line of // 

position, and the place of observation is ~r4r I " 

at some point of A B, or A B produced. Fig. 38. 

This is all which can be determined from an observed alti- 
tude, unless either the latitude, or the longitude, is definitely 
known. And as these are both uncertain at sea, except at 
the time when found directly by observation, the position 
of the ship found from a single altitude, or set of altitudes, 
is a line, of greater or less extent as the latitude, or the 
longitude, is more or less accurately known. 

In uncertain currents, or when no observations have been 
had for several days, the extent of this line may be very 
great. Yet, if it is parallel to the coast, it assures the na- 
vigator of his distance from land ; if directed toward some 
point of the coast, it gives the bearing of that point. 

225. If there is uncertainty in the altitude, for instance 
of 3', the line of position having been computed and plotted, 
parallels to it on each side may be drawn at the distance 
of 3'. 

So, also, if there is uncertainty in the Greenwich time, 
parallels may be drawn at a distance in longitude equal to 
the amount of uncertainty. 

In either case, the position of the ship is within the in- 
closed belt. 

In Fig. 38, a b is such a parallel to the line of position 
A B, its perpendicular distance from it measuring a differ- 
ence of altitude; the distance A a on a parallel of latitude 
measuring a difference of longitude. 



224: NAVIGATION. 

226. Since the line of position is at right angles with the 
direction of the body (Art. 222), the nearer the body is to 
the meridian in azimuth, the more nearly the line of position 
coincides with a parallel of latitude ; and thus a position of 
the body near the meridian is favorable for finding the lati- 
tude from an observed altitude, and not the longitude. 

So also, the nearer the body is to the prime vertical, the 
more nearly the line of position coincides w^ith a meridian, 
and the less does any error in the assumed latitude affect 
the longitude computed from an observed altitude. So 
that, if the body is on the prime vertical, a very large error 
in the assumed latitude will not sensibly affect the result. 
Such a position of the body is, then, the most favorable for 
finding the longitude from an observed altitude. 

These conclusions have been previously stated, drawn 
from analytical considerations. 

227. Two or more points of a line of position as (L Y A x ), 
(X 2 , A 2 ) etc., having been determined by Prob. 55, if the 
true latitude, X, be subsequently found, -p t> 



the corresponding longitude, A, may be 
obtained by interpolation. 

Or, the place of the ship may be 
found graphically upon the chart, by 



drawing a parallel in the latitude, i, -^ E Gr 
and taking its intersection P, with the 
line of position A B. 

So also, if the true longitude, /I, is subsequently found, 
the corresponding latitude, i, may be obtained by interpo- 
lation ; or, a meridian E F may be drawn in the longitude, 
A, which will intersect the line of position in P, the place 
of the ship. 

If there is uncertainty in either of these elements, two 
parallels of latitude (as in Fig. 38), or two meridians, may 
be drawn at a distance apart equal to the uncertainty. 



CIRCLES OF EQUAL ALTITUDE. 225 

As altitudes, latitudes, and longitudes are never found at 
sea with much precision, and may under unfavorable circum- 
stances be largely in error, the position of the ship on the 
chart is not properly a point, but a belt, more or less limited 
according to the accuracy of the elements from which it 
has been formed. 

228. In Fig. 39, if A is the position (i 19 A x ), 
B, the position (i 2 , A 3 ), 
both near P, the true position, whose latitude is 
i, and longitude is X ; 

the right triangles* A C B, A E P, being formed, 
CB = L 2 —L^ the difference of the two latitudes, 
AC = X 2 —X^ the difference of t?ie corresponding longi- 
tudes, 
E P = A L = L— i 15 the correction of X x , 
AE = JA = /l— A 1? the correction of X x ; then 
CB:EP = AC : AE 
or, (Z 2 — Z L ) : A L — (X 2 — A x ) : A A, 

whence we have, 

'*-«■£=£ } (162) 

and X — A x + A X ) 

as the formulas for finding A, the longitude of the true posi- 
tion, when its latitude, Z, is known. 
Or, we have 

** = "tt\ (163) 

and L — L^AL ) 

as the formulas for finding Z, when X is given. They are 
the same formulas as for an interpolation. The several 
differences are most conveniently expressed in minutes of 
arc, or, in the case of longitudes, in seconds of time. The 

* This is different from the projection on a Mereator's chart, where G B 
and E P would be augmented differences of latitude. 



226 NAVIGATION. 

local times may be used instead of the longitudes and in- 
terpolated in the same way. 

From the first of (162) we may readily determine how 
much a supposed error in an assumed latitude affects the 
resulting local time, or longitude. 

229. Problem 56. To find from a line of position the 
azimuth of the body observed. 

Solution. We have the positions (X n A x ), (X 2 ^ ^2)5 or the 
latitudes and longitudes of two points, from which the azi- 
muth, or course of the line of position, can be found by 
middle latitude sailing. 

Adding or subtracting 90°, according as the azimuth of 
the body is greater or less, gives the azimuth required. 

Or, a perpendicular to the line of position may be drawn 
upon the chart, and the angle which it makes with a meri- 
dian may be measured with a protractor. The azimuth 
may thus be found to the nearest 1°. 

Example. 

At sea, 1 865, Nov. 23, 10 J A. M., by account in lat. 36° 50'N"., 
long. 65° 20' W. ; Greenwich mean time 2 h 40 m 47 s ; the 
sun's correct central altitude 29° 6' 25" ; to find the line of 
position. 
G. m. t. Nov. 23 2 L 40 m 47 s =2\68 's dec. Eq'n of t 



Eq. of t. 


+ 13 18 


—20° 25' 33"— 30".9 


— 13 m 20 8 .4 + s .72 


G. ap. t. 


2 54 5 


(61 .8 


+ 1 .9 ( 1 .4 
—13 18 5 ( -5 






-1 23 -J 18 ,1 






— 20 26 56 ( 2 A 





1. With assumed Latitudes. (Prob. 43.) 

h r= 29° 6' 25" L x = 36° 30', Z 2 = 36° 50', Z3 = 37° 10', 



£1 = 36 30 1. sec 0.09482 .09670 .09861 

p— 110 26 56 1. cosec 0.02827 .02827 .02827 

2s= 176 3 21 

£-88 140 1. cos 8.53674 .49841 .45636 

s-h — 58 55 15 1. sin 9.93271 .93346 .93422 

G. ap. t. 2 h 54 m 5* 18.59254 .55684 .51746 



CIRCLES OF EQUAL ALTITUDE. 



227 



f (1) 22 28 37 1. sin 9.29677 


.27842 


.25873 


L. ap. t 

- 


I (2) 22 32 27 
( (3) 22 36 22 

f Ai = 4 25 28 = 66°22'.0 W. 

— 57'.5 
%z = 4 21 38 = 65 24.5 

— 58'.8 
Xs = 4 17 43 = 64 25.7 


L x - 36° 30' N/ 
U — 36 50 
Xg= 37 10 


r 



For JZ = +40', AX—— 116'.3 ; or a change of 40' in 
latitude produces a change of —116' in longitude. 

From A L — + 40', J A = — 116'.3, we find, by middle lati- 
tude sailing, the dep. 93'.0, and then the bearing of the line 
of position, regarded as a rhumb line, which it nearly is, 
NT. 66°.7 E. ; the sun's azimuth therefore is ST. 156°.7 E. 

Suppose the correct latitude to be 36° 57' X., to find the 
corresponding longitude on the line of position, we have 



; 36° 57' N. 



A-3 /u2 



Lz — L2 



Z 2 — 36° 50' N. h = 65° 24'.5 W. 

AL- +7' A% = — 2'.9x7= — 20'.6 

58'.8 
20 



= -2\94 



I =a 65° 4' W. 



2. W%A assumed Longitudes. (Prob. 46.) 



G. ap. time 


2 h 54 m 5 9 








&, = 4 19 3* 


— 411 21 m 3 


A 3 = 4 h 23 m 9 


L. ap. time 


* = — 1 24 55 


- 1 26 55 


— 1 28 55 




1. sec t 0.03052 


.03201 


.03354 


d-_20° 26' 56" 


1. tan d 9.57155 n 


.57155 


.57l55?i 




1. tanf 9.60207 w 


.60356?i 


.60509 71 




f =— 21° 48'.1 


-21° 52'.2 


-21°56'.4 




1. sin 9.56984 n 


.57114*1 


.57244 ?? 




1. cosec <# 0.45671 n 


.45671 n 


.45671 n 


h = 29° 6' 25" 


1. sin A, 9.68703 


.68703 


.68703 




1. cos 6' 9.71358 


.71488 


.71618 




f=+ 58° 51'.7 


+ 58° 45'.5 


+ 58° 89'.2 



S U = 37° 3'.6 IS T . Zs = 36° 53'.3 N. U = 36° 42'.8 K 
} 2,i = 64 45 W. A» = 65 15 W. A 3 = 65 45 W. 



228 



NAVIGATION". 



For A X = + 60', A Z = — 20'.8. From these, the bearing 
of the line of position is N. 66°.5 E. 

If the correct longitude is 65° 4' W., to find the corre- 
sponding longitude on the line of position, we have 



-k = 65° 4' W. 



Z 2 



h = 64° 45' W. Z, = 37° 3'.6 N. 

Al= +19' JL=— 0\34xl9 = — 6'.5 



Zi 10'.3 

AT3^ = -ij<r= - ' 34 



Z = 36° 57' N. 



Two assumed latitudes, or longitudes, would have sufficed, 
as the altitude is so small. 

230. Problem 57. To find the position of the observer 
from two altitudes of the same or different bodies, the Green- 
wich time being known. 

Solution. Find the line of position from each. If the lines 
are plotted on the chart, their intersection gives the position 
required ; as the lines A B, C D, in Figs. 40 and 41, which 
intersect in P. 

This intersection may also be readily found by computa- 
tion, when the lines are regarded as straight. 
Let L x and Z 2 be the same assumed 
latitudes in both computations ; 
and in Figs. 40 and 41, 

A, the point (Z 1? A' x ), called the 
first position, 

B, the point (X 2 , A' 2 ), 

both derived from the first ob- 
servation ; 

C, the point (i 1? X'\), 

D, the point (Z 2 , /l^), 

both derived from the second 
observation : 

the upper accents distinguishing the observations, the lower 
accents distinguishing the latitude used for each point. 




CIRCLES OF EQUAL ALTITUDE. 229 

P is the point of intersection, whose latitude L and longi- 
tude A are required. 

The diagrams are supposed to be constructed with the 
differences of latitude and longitude, so that in each 

ACz: X\— X n the diff. of long, in the lat. L u 
BD = ?J' 2 —X' 2 , the diff. of long, in the lat. L^ 
AE = J/l = A- A' 15 the correction of A' 15 which it is conve- 
nient to call the first longitude, 
EF = i 2 — i x , the difference of the assumed latitudes, 
EP = Ji = Z- i 15 the correction of the first latitude. 

In Fig. 40, B D lies in the same direction as A C, 

F P in the same direction as E F and E P ; and 

EF = EP-FP. 
In Fig. 41, B D is in the opposite direction to A C, and 

F P in the opposite direction to E F and E P ; 

and are therefore to be regarded as negative. 

We shall then have, algebraically, in both figures, 

EP = EP-FP. 

In the similar triangles ACP, B D P, 

AC:BD = EP:FP, 

and, by division, 

AC-BD:AC = EP-FP:EP = EF:EP, 

__ ACxEF 
whence EP = AO-BD ' 

or 

(*'l-*\)-(*%-*tf 

By (162) AX=AZ^£; 

or substituting for A L 

A -(/." 1 -A' 1 )-(^-A' a )- 



230 NAVIGATION. 

Patting m = — -|^--_, 

we have 

AL — m (Z 2 —Z l )^ 
A X = m {X' 2 — X\) 9 

Z = Z, + AZ, 

X = X\ + AX, 



(164) 



by which the latitude, X, and the longitude, X, of the inter- 
section can be found. 

231. Either assumed latitude may be designated as X x , 
and either observation by the accent ' , or be called the first 
latitude and the first observation ; but the several differences 
of latitude and longitude must be marked with their appro- 
priate names, or signs. 

If the differences of longitude X\ — X\, X\—X' 2y on the 
two parallels have the same name, their difference is taken 
in finding m, which will be +, when X'\—X\ > X\ — A' 2 , or 
the difference of longitude on the first parallel is the greater. 
In this case m > 1, A Z > Z 2 —Z l and A X > X f 2 —X\. The 
point P is then, as in Fig. 40, in the same direction as B 
from A, and beyond B. But m will be — , when X\—X\ and 
X\—X' 2 have the same name, and X\—X" l < X\—X\, or the 
difference of longitude on the first parallel, is the less. A Z 
and A X will then have different names respectively from 
Z, 2 —Z l and X r 2 — X\. In this case P and B are in opposite 
directions from A. A negative value of m may be avoid- 
ed, so that P and B will fall always on the same side of A, 
or P and D always on the same side of C, (Fig. 40), if toe 
take as Z x the latitude of the parallel on ichich is the greatest 
difference of longitude. 

If the differences of longitude, X\—X\, X ,r 2 —X' 2 , on the two 
parallels have different names, their sum is taken numerically 
in finding m / in that case m is -f and less than ], 
Ji< Z 2 —Z L and A X < X r 2 — X\, with the same names 



CIRCLES OF EQUAL ALTITUDE. 231 

respectively; and, as in Fig. 41, P is between A and B, and 
between C and D. 

When three or more latitudes are used in the computa- 
tions, those for which the differences of longitude are small- 
est should be taken as L x and X 2 . 

232. The more nearly perpendicular the lines of position 
are to each other, the better is the determination of their 
intersection. Hence, the nearer the difference of azimuths 
of the body or bodies at the two observations is to 90°, 
the better is the determination of position from double 
altitudes. 

If the azimuths are the same, or differ 180°, the two lines 
of ]30sition coincide in direction, and there is no intersection. 
In this case the great circle joining the two bodies, or the 
two positions of the same body, is an azimuth circle, and 
passes through the zenith. An approach to this condition 
is generally to be avoided. (Bowd., pp. 181, 195, notes.) 
Still, however, if the two bodies, or positions of the same 
body, are near the meridian, the lines of position nearly 
coincide with a parallel of latitude. The latitude is then 
well determined, but not the longitude. If the two 
bodies, or positions of the same body, are near the prime 
vertical, the lines of position more nearly coincide with a 
meridian and the longitude is well determined, but not 
the latitude. 

When the difference of azimuths is small, the intersection 
of the two lines may be computed with tolerable accuracy, 
while it cannot be definitely found by the projection of the 
lines upon a chart. 

233. The operations indicated in (164) are to subtract, 

1. The first assumed latitude from the second, {JL^—L^) ; 

2. for the first observation, the longitude corresponding 

to the first latitude from that corresponding to the 
second latitude, (A' 2 — A\) ; 



232 NAVIGATION. 

3. for each latitude, the longitude deduced from the 

first observation from the longitude deduced from 
the second, (A^ — X\) and (A" 2 — A' 2 ) ; 

4. the difference of longitude for the second latitude 

from that from the first, \_{a\— X\) — (A" 2 — A' 2 )], 
(or add numerically these differences of longitude when 
they are of different names.) 
Then 

5. To divide by this last result the difference of longi- 

tude, {h\— X\), for the first latitude, to obtain the 
coefficient m, (which will be — only when the dif- 
ference of longitude, (A /r 2 — A' 2 ), for the second lati- 
tude has the same name as and is greater than the 
difference of longitude, (A^ — A\), for the first lati- 
tude), 

6. To multiply m by the difference, (X 2 — Xj), of the 

two assumed latitudes to obtain the correction of 
the first latitude L x \ and by the difference, A' 2 — A^), 
of the two longitudes derived from the first ob- 
servation, to obtain the correction of the first of 
these longitudes, X\. 
These corrections have the same name as the differences 
from which they are derived, when ni is + ; but contrary 
names when m is — ; and are to be applied accordingly. 

234. The lines of position may be found from two as- 
sumed longitudes A L and A 2 , instead of two latitudes (Art. 
224). The formulas for finding their intersection will differ 
from (164) only by an interchange of the letters L and A. 
We shall have, then, 



m 



{L\-L\)-(L\-L'^ 

AX — ni! (A. 2 — A x ), A = l x + A A 

A L = m' (i. 2 -X , 1 ), L = L\ + AL 



(165) 



CIRCLES OF EQUAL ALTITUDE. 233 

Examples. 

1. With Xi = 30° 28' N"., Z 2 = 30° 8' K Diff. 20' S. 

by 1st alt. X\ = 59 15 W., X 2 = 59 W. " 15 E. 

by 2d alt. W\ = 58 43 W., // 2 = 59 8 W. " 25 W. 

Differences, 32~ E. — 8 W. = 40 E. 

32 

m = — = .8 

40 

J Z = — 20' x .8 = — 16', L — 30° 28' — 16' = 30° 12' ]S T . 

A A=r — 15' x .8 = — 12', 1 = 59 15 — 12' = 59 3 W. 

The differences of longitude on the two parallels, 32' E. 
and 8' W., being in opposite directions, the intersection is 
between the two parallels, or L is between i x and X 2 . 



2. With 


Li = 48° 10' S., 


L 2 = 48° 30' S. 


Diff. 20' S. 


by 1st alt. 


X'i = 88 16 E., 


a' 2 == 88 24 E. 


u 8 E. 


by 2d alt. 


?."l = 88 30 E., 


1\ = 88 55 E. 


" 25 E. 


Differences, 


14 E., 


— 31 E. 


= 17 W. 


m = — 


• 4 = -82 
17 






AZ = ~ 


-.82x20'~ — 16', 


Z = 4S° 10' — 16' 


= 47° 54' S. 


A ?> = - 


..82 x 8'=— 7', 


X = 88 16 — 7' 


= 88 9 E. 



In this example it is convenient to regard south latitudes 
and east longitudes as + . The differences of longitude on 
the two parallels, 14' E. and 31' E., being in the same direc- 
tion, the intersection is outside of the parallels and nearer 
the first, for which we have the smallest difference. 



3. With 


A, = 165° 


50' W., 


2* - 186° 20' W. 


Diff. 30' W. 


by 1st alt. 


L\ — 36 


16 S., 


X' 2 = 36 25 S. 


" 9 S. 


by 2d alt. 


L\ = 36 


38 S., 


L\ — 36 29 S. 


" 9 N. 


Differences, 




22 S, 


— "~T s. 


= 18 S. 


m' = 22 
18 


= 1.17 









JZ=l.l7x 9' = + 10'.5, L— 36° 16' + 10'= 36° 26' S. 
A i— 1.17 x 30' = + 35'.1, \ = 165 50 + 35 y = 166 26 W. 

The differences of latitude on the two meridians, 22' S 
and 4' S., are in the same direction ; and the intersection is 



234 



NAVIGATION. 



outside of the meridians and nearer the second, on which 
the difference of latitude is least. 

235. Problem 57 supposes the two altitudes observed at 
the same place. This at sea is rarely the case. 

Pkoblem 58. To reduce an observed altitude for a change 
of position of the observer. 

Solution. Let 
Z (Fig. 42) be the zenith of the place of observation ; 
h — 90° — Zm, the observed altitude; 
Z', the zenith of the new position ; 
h' = 90°— Z' m, the altitude reduced to the new position, Z f . 

d = Z Z', the distance of the 
two places, here referred 
to the celestial sphere ; 

C=FZ Z, the course ; 
Z = PZ m, the azimuth of m ; 
Z— C = m Z Z', the difference of 
the course and azimuth. 

Z Z', being small, may be regarded 

as a right line, 
Z 71 O as a plane right triangle, 
and O m, without material error, as equal to Z' m; so that 
we shall have 

Z O = Z Z' cos Z Z' m 
71 m = Z m - Z O 
or, putting J /i = Z O, 

J A = c?cos(C r -Z)) , v 

A h = d cos (C—Z) is, then, the reduction of the ob- 
served altitude to the new position of the observer : it is 
additive when C—Z< 90° numerically ; subtractive when 
C—Z > 90°. (Bowd., p. 183.) It is smaller, and can, 




CIRCLES OF EQUAL ALTITUDE. 235 

therefore, be more accurately computed the nearer C—Z 
approaches 90°. It is, therefore, better to reduce that alti- 
tude for which the difference of the course and azimuth is 
nearest 90°. 

If the second is the one reduced, then C is the opposite of 
the course. 

In practice Z Z' does not usually exceed 30', so that al- 
though an arc of a great circle of the celestial sphere, it may 
be regarded as representing the distance, c?, of the two 
places on the earth ; or, at sea, the distance run. The azi- 
muth, or bearing, of the body can be observed with a com- 
pass, or be computed to the nearest degree, or half-degree, 
from the altitude. 

The assumption, 71 m = O m, is more nearly correct, the 
greater Z' m or Z m, that is, the smaller the altitude. If 
we treat Z Z' m as a spherical triangle, d = Z Z' being ex- 
pressed in minutes and still very small, we shall find 

A h = d cos (C-Z)-i cP sin 1' tan h sin 2 {C—Z); (167) 

but the last term is inconsiderable unless d and h are both 
large. For instance, if d = 30', it will not exceed 1' unless 
h > 82°. 

Example. 

The two altitudes of the sun are 36° 16' 20", 58° 15' 20% 
the compass bearings of the sun respectively S. E. by E. -J E. 
and W. S. W. ; the ship's compass course, and distance 
made good in the interval N". N. W". \ W. 25 miles ; 

S. f| E. differs from N". 2\ W. 13 points, so that the re- 
duction of the 1st altitude to the position of the 2d is 

25' X cos 13 pts. == — 25' cos 3 pts. = — 20'.8 = — 20' 48". 

S. 6 W. differs from S. 2| E. 8£ points, and the reduction 
of the 2d altitude to the position of the 1st is 



236 NAVIGATION. 

25' cos 8| pts. s= — 25' cos 7£ pts. = — 2' 30"; 
or — 2' 39", if the last term of (167) is included. 

236. By (166) or (167) we may reduce one of the two alti- 
tudes for the change of the ship's position in the interval. 
But instead of this we may put down the line of position 
for each observation, and afterwards move one of them to a 
parallel position determined by the course and distance 
sailed in the interval. Thus in Fig. 43, let 

2> A B be the line of position for the 

first observation, and 




A a represent in direction and length 

the course and distance sailed 

Fig 4a in the interval ; then 

a b, drawn parallel to A B, is the line of position which 
would have been found had the first altitude been observed 
at the place of the second. 

If the second observation is to be reduced to the place of 
the first, then A a in direction must be the opposite of the 
course. 

The perpendicular distance of A B and a b is the reduc- 
tion of the altitude for the change of position: for that dis- 
tance is A a X cos (B A a — 90°). 

LATITUDE BY TWO ALTITUDES. 

237. Iii Sumner's method the latitude and longitude are 
both found by two altitudes, either of the same or different 
bodies. It is sometimes desirable to find the latitude only, 
or at least to make this the chief object of combining the 
two observations. 

238. Problem 59. To find the latitude from tioo altitudes 
of the sun, or other body, supposing the declination to be the 



LATITUDE RY TWO ALTITUDES. 



287 



same at both observations, and the Greenwich time to be 
known approximately \ 

Solution. Let two altitudes, or sets of altitudes, be ob- 
served and the times noted by a chronometer, or a watch 
compared with it; reduce the altitudes to true altitudes, and 
at sea one of them for the change of the ship's position in 
the interval by Prob. 58. Find also the difference of the 
chronometer times of the two observations, and correct it 
for the rate in the interval. 

t Ac 
This correction is -^— (+ when the chronometer loses), 

t being the interval in hours of chro. time, and 
A c the daily change. 

The result is the elapsed mean time for a mean time chro- 
nometer ; the elapsed sidereal time for a sidereal chrono- 
meter. 

The Greenwich mean time of the greater altitude, or of 
the middle instant, should also be obtained from the chro- 
nometer times, sufficiently near for finding the declination 
of the body. 

In Fig. 44, let 
M and M' be the two positions of 

the body, 
h — 90° — Z M, the first altitude, 
ti= 90°— Z M', the second altitude, 
d = 90°- P M = 90° - P M', the 

common declination, and 
t = M P M', the difference of the 
hour-angles, 
Z P M and Z P M', of the body 
at the two observations : 
or, letting Tand T designate these hour-angles in the order 

of time, 

t = T f -T. 




238 NAVIGATION. 

The method for finding t is different for different bodies. 

a. For a fixed star the angle M P M', or t, is the elapsed 
sidereal time. An elapsed mean time must therefore be re- 
duced to the equivalent sidereal interval. If, then, 

S and S' represent the sidereal times of the two observations, 

t m and s, the elapsed mean and sidereal times, 

we have, when the sidereal interval has been found, 

t=S*-S.= s; (168) 

when the mean time interval has been found, by (85), 

t = t m + . 00214 t m ; (169) 

or, with t m expressed in hours in the last term (87), 

t = t m + 9 S .8565 t m . (110) 

b. For the moon or a planet, if 

a and a -f A a represent the right ascensions of the body at 
the two times, we have (Art. Ill), 

T=S-a, T f =S'~ a -A a, 

and 

« = f-r=«-jc, (171) 

that is, the elapsed sidereal time diminished by the increase 
of the right ascension of the body in the interval. 

A h a, the change of right ascension in l h of mean time, 
may be obtained from the Almanac for the middle Green- 
wich time. 

The change in l h of sidereal time will be, by (86), 
(1— .00273) 4*j 
which can readily be found by regarding A h a as a sidereal 
interval, and reducing it to its equivalent mean time inter- 
val. 

Expressing t m and s in hours, when used as coefficients, 

Ave have 

Aa — t m . A h a = s. A h a (1— .00273) ; (172) 

and, for an elapsed sidereal time, 

t = 8—.s. A h a (1— .00273) ; (173) 



LATITUDE BY TWO ALTITUDES. 



239 



for an elapsed mean time, by (170), 

t = t m + t m (Qs.8565 -A h a). (174) 

c. For the sun, the angle MP IT, or t % is the elapsed ap- 
parent time. If, then, A h JS is the hourly change of the equa- 
tion of time (+ when the equation of time is additive to 
mean time and increasing, or sub tractive from mean time 
and decreasing ; that is, when apparent time is gaining on 
mean time), 

t = t m +t m . A h E, (175) 

by which t may be found from a mean time interval. If the 
sidereal interval is given, we have, as in (173), for a planet, 
t = s — s. A h a (1 — .00273). 

The reduction of the elapsed mean time to an apparent 
time interval, is commonly neglected by navigators; but on 
December 21, A h E= l s .25, and during a large part of the 
year exceeds s . 5. 

239. We have given 

h = 90°- Z M, d = 90°- P M = 90°- P M', 

A'=90°-ZM', * = MPM', 

to find from the several triangles of Fig. 44, the latitude, 
Z=90°-PZ. 
Various solutions of this problem have been given, from 
which the following are selected. 

240. (A.) 1st Method hi BowdiicNs Navigator. 

Let T (Fig. 44), be the middle point of M M' ; join P T and 

Z T ; and put 
4 = MT = M'T=JM M', 

or half the distance of the 

two positions of the body, 
B — 90°— P T, the declination 

of T, 

j?=:90 o -Z T, the altitude 
of T, 

J = PTZ, the position angle 




of T. 



Fig. 44. 



240 NAVIGATION. 

AsPTM and P T M' are equal right triangles, we have the 
angle PTM = PT W= 90°, 

2 = 90°- Z T M = Z T M'— 90°, 
and J« = MPT = M'PT. 

1. In the right triangle PTM, by Sph. Trig. (80), (84), 
and (82), 

sin A = cos d sin -J- Z, ) / ^ x 

sin B = sin d sec JLy ) 
or tan B == tan c? sec -J £/ 0-^) 

by which ^4 and .S, or M T and 90°— P T, can be found. 

2. In the two triangles Z M T, Z M' T, by Sph. Trig. (4), 

sin h = sin H cos A + cos H sin A sin 97 
sin h f = sin .Z? cos A— cos .Z? sin .4 sin q ; 
the half difference and half sum of which, by PI. Trig. (106) 
and (105), are 

sin \ {h—ti) cos \ (h-{-h f ) = cos J? sin A sin q, 
cos -J- (h—h') sin f (h + ti) = sin J? cos ^ty 
from which, 

COS J. v 7 

sin a = sini(A-/Qcosi(A + ^) fl ^ 

^ cos H sin J. ' v ' 

which determined and y, or 90°— Z T and the angle P T Z. 

3. In the triangle PTZ, by Sph. Trig. (4), 

sin L = sin B sin II + cos B cos H cos #, 

To adapt this to computation by logarithms, put 

cos C sin Z= cos J? cos ^, | /180'i 

cos C cos Z = sin H y ) 

and then 

tan Z = cot iZ cos q, ) 

cos 6 7 = sin II sec Z, > (181) 

sin L = cos (7 sin (jS + Z), ) 



LATITUDE BY TWO ALTITUDES. 241 

which determine (7, Z, and L. If, however, we add the 
squares of (180), we shall have 

cos 2 C = cos 2 H cos 2 q + sin 2 H, 
or 1 — sin 2 C — cos 2 H (1 — sin 2 q) + sin 2 H; 

whence 

sin (7= cos J? sin q. (182) 

Substituting this in (179), and the 2d of (180) in (178), we 

have 

. n sin \ (Ji — h') cos \ (h + h') 

Sill C — ; z « 

sin J. 



~ cos i Qi—li') sin |- (h + h') 

COS ^ — ■: -pz , 

cos A cos 6 
and (181) sin i = cos C sin {B + Z). 



(183) 



J£, being found from its cosine, may have two values 

numerically equal with contrary signs. Representing these, 

we have 

sin L = cos C sin (B±Z), 

which gives two values of X. The value which accords 
most nearly with the latitude by account may be taken. 
We shall see presently how the admissible value of Z may 
be selected. 

241. To avoid using both the sine and cosecant of A and 
the cosine and secant of (7, we may take the reciprocals of 
(176) and the 2d of (183) ; we shall then have, as in the 1st 
method of Bowditch (p. 180), 

cosec A = sec d cosec* \ £, 1 

cosec B = cosec d cos A, 

sin C = sin % (h—ti) cos $ (h + ti) cosec A, > (184) 

sec Z— sec \ (h—ti) cosec £ (h + ti) cos ^4 cos (7, 
sin L = cos C sin (J5±Z). J 

* log sec -J tf and log cosec -J ^ may be taken from Table XXVII. corre- 
sponding to t in the column P M (Art. 126). 



242 



NAVIGATION. 



It is unnecessary to find A and (7, as log cosine A can be 
taken from the tables corresponding to log cosecant A, and 
log cosine C corresponding to log sine C Indeed, we may 
dispense with A entirely by substituting the 1st of the pre- 
ceding equations in the 3d, and the 2d in the 4th, and em- 
ploying (177). We shall then have, for the solution of the 
problem,— 



tani?= 



:tan d sec* -J £, 



(185) 



sin C— sin \ (h—h f ) cos J (h + ti) sec d cosec* ^ £, 
sec Z=sec| {h—ti) cosec \ (]i + h!) sin d cosec B cos 6 r , 
sin i=cos G sin (B±Z). 

A, B, C, Z, and i, are each numerically less than 90°; 

A is in the 1st quadrant; 

C is + when the 1st altitude is the greater, — when it is 

the smaller ; 
B has the same sign, or name, as the declination ; and 
L the same as (B + Z) or (B— Z),from which it is obtained. 

242. If Z O (Figs. 44 and 45), be drawn perpendicular to 
P T, we shall find from (182), 

C=±Z O, + when P T is west, 
— when it is east, of the meri- 
dian ; Z=T O, 

B + Z=90°-P O in Fig. 44, 

JB-Z=90°-P O in Fig. 45 ; 

Z, or T O, being -f or — according 
as P and Z are on the same side of 
M M', as in Fig. 44, or on opposite 
sides, as in Fig. 45. 

This may also be shown in an- 
other way : for, in the first case, the 




* log sec i t and log cosec i t may be taken from Table XXVII. corre- 
sponding to t in the column P M (Art. 126). 



LATITUDE BY TWO ALTITUDES. 



243 




angle q = P T Z < 90°, cos q is + ; 
and since, in (180), cos C and sin JI 
are -f , sin Z, and therefore Z, will 
also be + . In the second case, when 
M M' produced passes between P 
and Z, q = P Z Z > 90°, cos q and, 
consequently, sin Z and Z are — . 

Instead of marking Z + or — , we 
may use the symbols N" and S, as 
for c?, _Z>, and X. We shall then 
have the rule (Bowd,, p. 181) : — 

Mark Z north, or south, according as the zenith and north 
pole, or the zenith and south pole, are on the same side of 
the great circle, which joins the two positions of the body. 
By thus noting the position of this circle, the ambiguity of 
Z is removed. 

We may, however, remove the ambiguity by noting the 
azimuths of the two points M and M'. 

In Fig. 44, P Z M > P Z M'; in Fig. 45, P Z M < P Z M'; 
which would be the case also if one or both points were on 
the other side of the meridian. Hence we have the rule : — 

Z has the same name as the latitude when the azimuth of 
the body is numerically the greater at the greater altitude ; 
but a different name from the latitude when the azimuth at 
the greater altitude is the less. The azimuths are to be 
reckoned both east and west from to 180°, and from the 
N". point in north latitude ; but from the S. point in south 
latitude. 

243. If Z is very small, it cannot be accurately found from 
its cosine, or secant ; its sign may be doubtful ; and the lati- 
tude cannot be determined with precision. This will be the 
case, when the altitudes are very great ; when M and M' are 
near the prime vertical ; or, in general, when M and M' are 
remote from the meridian and the difference of azimuths, 



244 NAVIGATION. 

M Z M', is either very small, or near 1 80°. In each of these 
cases M M' intersects the meridian very near the zenith. 

It has been seen, with regard to lines of position derived 
from two altitudes (Art. 232), that the most favorable con- 
dition is when M Z M'=90° ; but that the latitude alone can 
be well determined when M and M' are quite near the meri- 
dian in azimuth and M Z M' quite small. Indeed, if both 
azimuths are 0, or 180°, the two altitudes become a meridian 
altitude. 

These conditions belong to all methods of finding the lati- 
tude from two altitudes. 

244. The latitude having been found, we may proceed to 
find the hour-angle of the body from one of the altitudes 
(Prob. 43), if it is sufficiently near the prime vertical, and 
thence the longitude, if the times have been carefully noted 
by a Greenwich chronometer (Art. 188). 

Instead of this, by putting T = \ (T+T') = Z P T, the 
middle hour-angle, we have the formula (146), 

cos a cos L sin \ t v ' 

and from (185) 

• n — sni 2- (h—h') cos -J- (h + h') 
sin — j — : — 7— 1 x 

cos a sin -J- t 

whence 

S i n *^ = !^. (187) 

cosZ v y 

This could also have been obtained from the right triangle 
P O Z, (Figs. 44, 45) ; from which we have also 

7DA tan Z 

tan ZrO — - — «a, 

sin P O 7 

or 

* If we enter Table XXVII. (Bowd.) with log sin T 01 or log tan T„ we 
shall find 2 T Q corresponding in the P. M. column. 



LATITUDE BY TWO ALTITUDES. 245 

Thus, by a brief additional computation,^ can be found 
by (186) or (187). We shall have also 

T=T Q -±t, T=T + it (189) 

for the hour-angles at the times of the observations. The 
longitude can be found from either T , T^ or T\ and the cor- 
responding chronometer time. 

(186) is the formula of Littrow's method* (Art. 200). 
The favorable conditions, as there stated, for finding 7J, are 
a small value of T and high altitudes near the meridian, or 
altitudes on each side of the meridian near the prime verti- 
cal. But such altitudes are unfavorable for finding the lati- 
tude. 

When both latitude and longitude are to be found from 
two altitudes, the nearer the difference of azimuths is to 90° 
the better will be the determination. The most favorable 
conditions for combining them will be equal azimuths of 45°, 
or 135°, on each side of the meridian. 

If one of the altitudes is very near the prime vertical, and 
the other very near the meridian, it will generally be better 
to find the time and longitude from the first by Prob. 43, and 
the latitude from the second by Prob. 46 or 47. 

245. In this problem the declinations are supposed to be 
the same at both observations. This will be the case with 
the sun only at the solstices ; with a planet, or the moon, 
only when 90° from its node, and with the latter body for a 
very brief period. Navigators usually neglect the change 
of declination of the sun, or a planet, and use the mean de- 
clination, or that for the middle instant. It is better, how- 
ever, when the change is neglected, to employ the declina- 
tion at the time of the greater altitude,f except when the 

* The novelty of Littrow's method consists in finding T^ from very high 
altitudes near the meridian. (146), or (186), is by no means a new formula, 
f Chauvenet's Astronomy, Yol. L, pp. 276, 315. 



246 NAVIGATION. 

hour-angle of this altitude is greater than the middle hour- 
angle. This can be the case only when the altitudes are on 
different sides of the meridian. When the middle declina- 
tion is used, we may, with little additional labor, find the 
correction of the computed latitude by the following formula 
from Chauvenet's Astronomy (Vol. I., p. 267). 

AZ = — ^: sin f, (190) 

cos L sin \ 1 v ' 

or, by substituting (187) 

JZ^-l^ZL (191) 

sin \ t v 7 

in which A d is half the change of declination in the inter- 
val of the observations. Noting whether it is toward the 
north or the south, we can apply it with the same name to 
the computed latitude, when the lesser altitude was ob- 
served first / but with a different name when the lesser alti- 
tude was observed last. 

With this correction the preceding method can be em- 
ployed for altitudes of the moon at sea, when the elapsed 
time does not exceed l h . 

The correction of T^ the middle hour-angle, may also be 
found by the formula (Chauvenet's Astronomy, Vol. I., 
p. 268,) 

AT - Ad ( taU L C ° S To - i md ) • (1921 

which differs from the equation of equal altitudes (136) only 
in the first term being multiplied by cos jT , and in chang- 
ing the signs, as it is here applied to the hour-angle instead 
of the chronometer time. 

(B.) Domoes^s Method ; JBowditcNs 2d Method: with an 
assumed latitude. 

246. This method differs from the preceding in first find- 
ing T^ the middle hour-angle, by using an approximate lati- 



LATITUDE BY TWO ALTITUDES. 



247 



tude, and then the latitude from the greater altitude and its 
computed hour-angle, as in Problem 46 for finding the lati- 
tude from a single altitude. 



N 




Letting U == the assumed latitude, 










and as before 


/ /^7l 


P 






T o =i (T r + T) = ZFT 


1 /°/J 








(Fig. 46) the 


f ' 


z 






middle hour- 


\^g. 








angle. 

*==} (f-r)=MPir, 

the difference of 


5. 


3 






the hour-angles, 


Fig. 46. 




we maj 


r use the formulas of Art. 244, 


sin 


c = 


sin \ (h — 
cos 


A')cosi-o+^r 

d sin ^ t 




sin 


T - 

•*■ 


sin 
cosZ 




(193) 


T= 


= T - 


~i t, T< = 


T + ht 




and of Art. 


149, 









(194) 



cos z = sin h + 2 cos rf cos J7 sin 2 £ T 7 
cos z Q = sin A'— 2 cos c? cos i' sin 2 -J- 7"" 

selecting that which contains the greater altitude and less 
hour-angle. 

z = 90°— h is the meridian zenith distance from which 
the latitude may be found as from an observed meridian 
altitude. (Prob. 45.) 

Should this differ from the assumed latitude, the computa- 
tions should be repeated, using this new value. 

The method can be used to most advantage when T Q is ' 
small, and the greater altitude is observed near the meri- 
dian. But it has the inconvenience of requiring several re- 
computations, when the computed latitude differs widely 
from that assumed. If the observations are unfavorable for 



248 NAVIGATION. 

finding the latitude, successive recomputations will approxi- 
mate very slowly, or may not approximate at all, to a con- 
clusive result. When the less altitude is near the prime ver- 
tical, it is preferable to find from it the greater hour-angle, 
and then the less by adding or subtracting t. 

247. Mr. Douwes's formulas, however, are somewhat dif- 
ferent. 

From the triangles Z P M, Z P M' we have by Sph. Trig. (4) 

sin h = sin d sin L + cos d cos X cos jP, 
sin h' = sin d sin i + cos d cos L cos T f / 

the difference of which is 

sin h — sin h! =s cos d cos L (cos T— cos T r ). 

This by PI. Trig. (130) becomes 

sin A— sin h'= 2 cos d cos L sin f (T+ T 7 ) sin £ {T'-T) ; 

whence, since 

T = i (r+ r ) and i r = | (r- T 7 ), 
2 sin 2^ = (sin ti — sin A) sec d sec i cosec % £, (195) 

which gives T 0J provided for L we use the approximate la- 
titude L'. • 

We then have as before 

T=T-it, r= T + ±t (196) 

and for finding the meridian zenith distance from the greater 
altitude, (116) 

cos z = sin h + cos i' cos d versin T 7 , ) 
or, j. (197) 

cos Zn = sin A' + cos U cos c? versin T", ) 

Mr. Douwes prepared special tables (Tab. XXIII., Bowd.,) 
to facilitate the use of formulas (195) and (197) calling 
log cosec \ t, log of " half elapsed time," 

log (2 sin T ), log of "middle time," 

log versin t, " log rising " ; 



LATITUDE BY TWO ALTITUDES. 



249 



increasing the indices of the last two by 5, so that natural 
sines, etc., to 5 decimal places may be treated as whole 
numbers. 

(193) and (194) are preferable, as they require only tables, 
which are in common use. 

248. When the declination is taken for the middle time, 
we have for the correction of the computed latitude (191) 

A d. sin T 



AZ = 



sin 4 t 



(198) 



in which A d is the change of declination in half the inter- 
val of time between the observations. 



Example. 

At sea, 1865, April 16, A. M. and P. M., in lat. 37° 20' S., 
long. 12° 3' W., by account, two altitudes of the sun were 
observed for finding the latitude, the corrected times and 
altitudes being as follows : 



G. m. t. April 15 23 h 16 m 25 3 
" " 16 1 IS 26 



Mid.G.m.t. " 
— Eq. of t. 
Mid. G. ap. t. 

Elapsed m. t. 
Ch. of eq. of t. 
Elapsed ap. t. 



16 



17 25.5 
+ 16.4 
17 42 



O's true alt. 38° 16' 25" 

" 42 15 30, 

^(h + li)- 40 15 57 
£(£— h!)=z— 1 59 32 



S. 15£pts.E. 
S. 161 W. 



2 1 

+ 1 
2 2 



O's dec, d = + l0° 13' 57 ff 
Ch. in 1* +53.04 

Ch. in l h .0l7 J d = + 53.7 



Computation by (185), (187) and (191). 



1. sec \ t 
\. tan d 
1. tan B 



0.01558 
9.25651 

9.27209 



. .5 = + 10° 35' 52' 
Z-—4.1 45 25 
B+Z -—37 9 33 



1. cosec i t 0.58001 

1. sec d 0.00696 1. sin d 9.24955 

1. cos i(h+h') 9.88256 1. cosec \ (Ji+h') 0.18954 

1. sin i(h+h') 8.54113 n 1. sec|(/i— A') 0.00026 

1. sin C 9.01066 ?! 1. cos C 9.99771 

1. cos C 9.99771 1. cosec B 0.73539 

1. sin (B+Z) 9.78106 n 1. sec Z 0.17245 



250 



NAVIGATION. 



Xi =-36 55 52 1. sinZ 9.77877 t* 

AL— +26 mid. G.ap.t. = h l7 m 42 3 

L= 36 55 26 S. T ^-0 29 28 
Long. + 47 10 

or 11° 47' 30" W. 



1. sec L 

1. sin C 
1. sin 1\ 
1. cosec -J- 1 
log J d 



0.09726 

9.01066m 

9.10792n 

0.580 

1.730 



log(-JZ) 1.418 n 



IS 



The azimuth at the greater altitude being the greater, Z 
— , or S., like the latitude. 



Computation by (193), (194) an^ (191), with assumed lat. 

37° 20' 8. 



t = 2 h 2 E1 2 s 


L 


, cosec \ 0.58001 






i(h + h')= 40° 15' 57 


1. 


, cos 


9.88256 






£(A_A')=- 1 59 32 


1. 


sin 


8.54113n 


log 2 


0.30103 


d = + !0 13 57 


1. 
I. 


sec 
tan 


0.00696 


1. cos d 
1. 2 cos d 


9.99304 


C 


9.01066 n 


"O29407 


U =-37 20 


1. 


sec 


0.09957 


1. cosX' 


9.90043 


2;=— h 29 ra 37 s 


1. 


sin 


9.11023 n 


21. s'miT' 7.67078 


i*=+l 1 1 












T' =+0 31 24 






733 


log 


7.86528 


h' - 42° 15' 30" 






sin .67248 






z =-47 10 25 






cos .67981 






d= 10 13 57 












L x =-36 56 28 













2d Approximation. 



c 






1. tan 9.01066 n 




1. 2 cos d 


0.29407 


L x =-36° 


56' 


28" 


1. sec 0.09732 




1. cos L" 


9.90268 


T Q =— h 29* 


1 28 s 


1. sin 9.10798 n 








it = + 1 


1 


1 


sin/i' .67248 




2 1. sin i T 


7.67488 


T =+ 


31 


33 


744 




log 


7.87163 


z ft =-47 


9 


46 


cos .67992 




1. sin T 


9.108 n 


<Z = + 10 


13 


57 






1. cosec i t 


0.580 


X 2 =-36 


55 


49 Mid, 


.G.ap.t. =— h 17 nl 


L 42 8 


log A d 


1.730 


JX = 


+ 


26 


T =-0 29 


28 


log (- A L) 


1.418 71 


X= 36 


55 


23 S. 


Long. +0 47 
or 11°. 47' : 


10 

50" 


W. 





LATITUDE BY TWO ALTITUDES. 251 

(C.) Method of Equal Altitudes. 

249. When the altitudes are equal, or h=h\ the equations 
of (185) become, 

C=0 
tan B = tan d sec \ t 
sin h sin B 



COS Z = 



sin d 

Z = B ± z 



(199) 



which are identical with (113) for finding the latitude from 

a single altitude. 

We have also T = —\ t, T = \ % T = 0; and from (194) 

cos z — sin A 4-2 cos U cos d sin 2 \ t, (200) 

from which the meridian zenith distance may be found by 
one or more approximations. 

L having been found by using the declination for the time 
of meridian passage needs no correction for a small change 
-of declination, since in (198) sin T = 0. 

(D.) Chauvenef s Method, by two altitudes near the meri- 
dian w hen the time is not known. (Astronomy, Vol. I., 
p. 296.) 

250. The method of reducing circum-meridian altitudes 
to the meridian, when the time is known, has already been 
given, (Prob. 47). At sea, however, the local time is fre- 
quently uncertain ; while altitudes near the meridian are re- 
sorted to as next in importance to meridian altitudes for 
finding the latitude. 

As in Prob. 47, let h represent the meridian altitude, 

. . r.96349 cos L cos d . nm 

J h — = — TT 1\ 5 the change of altitude in l m 

sin (L—d) ' ° 

from the meridian, (Tab. XXXII., Bowd.,) 
and as before, 



252 NAVIGATION. 

h and h\ the true altitudes, 

j? 7 and T\ the corresponding hour-angles, (in minutes of 

time,) 
t = T'—T, the difference of the hour-angles, 
T = i (T'+T), the middle hour-angle. 

By (120) 

h = h+A h.T% ) , , 

The mean of these equations is 

h Q = t(h + h') + (T'*-l n ). A h. (202) 

But 

(T TV (T'+TV 

t\- p = (L-£) + (L+d.'j = {i t y + T a > 

which substituted in (202) gives 

h = i (h + K) + [i t *+T >] A h. (203) 

The difference of the two equations of (201) gives 
h-h' = (T*-T 2 ). A h = 2 T 1. A Q h. 



Hence 






Substituting this in (203) we have 

h = HM ) + (^J h + ^p^l. (205) 

The reduction to the meridian, then, is effected " by add- 
ing to the mean of the two altitudes two corrections; 1st, 
the quantity (| t) 2 . A A, which is nothing more than the 
common reduction to the meridian (120) computed with the 
half-elapsed time as the hour-angle; 2d, the square of one 
fourth the difference of the altitudes divided by the first 
correction." Several pairs of altitudes can be thus com- 
bined, and the mean of the meridian altitudes taken, from 
which the latitude can be obtained as from an observed me- 
ridian altitude. 



LATITUDE BY TWO ALTITUDES. 253 

251. The restriction of the method corresponds with that 
of circum-meridian altitudes (Art. 150).* Quite accurate 
results can be obtained with hour-angles limited to 5 m when 
the altitude is 80°, to 25 m when the altitude is only 10°. If 
the interval t, however, exceed 10 m , AJi should be computed 
to two or three places of decimals, as it is given in Table 
XXXII. (Bowd.) only to the nearest 0".l. 

The accuracy of the method depends mainly upon the ac- 
curacy of the 2d correction, and therefore upon the preci- 
sion with which the difference of altitudes has been obtained. 
The altitudes, then, should be observed with great care. 
Errors of the tabulated dip and refraction, and a constant 
error of the instrument will affect both altitudes nearly alike. 

* Xote to Art. 150 (omitted in its proper place). 
From (117) we have with more exactness, 

cos L cos d 2 sin 2 4- t 



Ah: 



or putting 



sin (L—d) ' sin 1" 



2 sin 2 i t , . cos L cos d 

m = — : — —j— and A = — — jj — jt-, 
sm 1 sm (L—dy 

Ah = Am and h = h -\-A m. 
Delambre's formula, obtained by developing in series the preceding equa- 
tion, (116) is 

h = h -\-A m —B n, 
in which 

2 sin 4 !-* ,_ _ 

n = — , , £ = A 2 tan (Z— d) 

Table Y., of Chauvenefs Astronomy, contains m and ra, and Table VI. 
contains log m and log n for different values of t from to 30 m . 

Table VII. A gives the hmiting hour-angle at which the error resulting 
from neglecting the 2d reduction, Bn, amounts to 1". It varies from in 
the zenith to 36 m in latitude 40°, or to 67 m in latitudes 0° and 80°, for an 
altitude of 10°. 

Table XXXII. (Bowd.) gives J h only to the nearest 0".l ; if, then, it is 
taken from this table, AJi. t 2 may be in error 1", if t > 4 m . If, however, 
AJi is computed to the nearest 0".001, the error of using J /i. f, instead of 
A m, will not exceed 1", unless t > 20 m and h > 60°. 



254 NAVIGATION. 

If the altitudes are equal, this second correction becomes 0. 
The most favorable condition is, therefore, that of equal alti- 
tudes observed on each side of the meridian. 

At sea, the method is especially useful for altitudes of the 
sun observed with a clear, distinct horizon. A long interval 
between the observations is to be avoided on account of the 
uncertainty of the reduction of one of the altitudes for the 
run of the ship. 

252. The hour-angle of either altitude may also be ob- 
tained approximately; for we have from (204), in minutes, 

(Ji-h') 



and 




(206) 



Example. 



1865, March 14, near noon, in lat. 45° 30' S., long. 120° 10' E.. 
by account, two altitudes were observed for latitude, 
T. by Chro. 4 h 15™ 20 s ; Obs'd alt. 46° 45' 30", (North) 
" " " 4 26 16 " " " 46 54 40; 

Index cor. of sextant +5' 20"; height of eye 18 feet. 
The sun's declination at noon —2° 31' 57", H. ch. +59". 
By preliminary computation AJi — 2".02 ; log AJi — 0.306. 
t = 10 m 56 3 h—h' = — 9' 10" 

i t = 5 28 = 5 m .47 l(h—h')=— 137 
(ity= 29.9 '$(h + h')= 46° 50' 5" 

Aji= 2 ".02 1st cor. 29.9 x2".02= + l 

137 2 



2 log i {h-h') 


4.274 


log (1st cor.) 


1.781 


log (2d cor.) 


2.493 


log i (h-h') 


2.137 n 


ar. co. log -} t 


9.262 


ar. co. log A Q h 9.694 


log T Q 


1.093 


T«=~ 


■ 12 m .4 


T = — 


■17 .9 


r=- 


■ 6 .9 



2d cor. -— — • 
60.4 




= +5 11 
46 56 16 


In. cor. 




+ 5 20 


S. diam. 




+ 16 7 


Dip 




— 4 11 


Ref. and par. 




— 48 




K = 


: 47 12 44 




z o — 


: 42 47 16 S. 


(12 m before h 


)d= 


: 2 32 9 S. 




Ls= 


45 19 25 S. 



LATITUDE BY TWO ALTITUDES. 255 

{JE) PresteVs method* by the rate of change of altitude 
near the prime vertical. 

253. In the note to Art. 197 we have, for a very brief in- 
terval of time, and a small change of altitude, 

A t - Alh 

15 cos L sin Z' i 

or, using the notation of this problem, 

T'-T=t = „ h '~ A . „ ; 
15 cos L sin z 7 

whence 

cos L = ~ cosec Z; (207) 

in which A'— h is expressed in seconds of arc, and t in 
seconds of time, and, Z being + when east, — when west, 
cos L is always positive. If Z is near 90°, its cosecant 
varies slowly. When Z = 90°, we have, 

cosZ=4^. (207') 

If, then, two altitudes are carefully observed near the 
prime vertical, and the times noted with great precision, the 
interval not exceeding 8 or 10 minutes, an approximate lati- 
tude may be found by (207 ; ), when the altitudes are within 
2° or 3° of the prime vertical; or by (207) when they are at 
a greater distance, and Z is approximately known. 

The time of passing the prime vertical can be found by 
(107). Z maybe roughly computed from the altitudes, or 
found within 2° from the bearing observed by a compass, 
which will suffice, if the observations are made within 10° 
of the prime vertical. 

As, near the prime vertical, the altitude changes uniformly 
with the time, several altitudes may be observed in quick 
succession and the mean taken as a single altitude. 

The larger h'—h and t, consistent with the supposition of 
uniformity of change and the condition by which they are 

* Chauvenet's Astronomy, I., pp. 303, 311. 



256 NAVIGATION. 

substituted for their trigonometric functions, the more accu- 
rate in general will be the result. 

Still the method does not admit of much precision. It 
is entirely unavailable near the equator, and in latitude 45° 
may give a result in error from 5 to 10 minutes, even 
when the greatest care has been bestowed on the observa- 
tions. It may, however, be useful to the navigator in high 
latitudes, as it can be used for altitudes of the sun, when it 
is almost exactly east or west, and consequently when no 
other method is practicable. There are occasions at sea, 
when to find the latitude only within 10' is very desirable. 

Examples. 

1. 1865, June 15, 7 h A. M., in lat. 60° N., Ion. 60° W. ; 
T. by Chro. ll h 13 m 25 s .3, obs'd alt. © 27° 0' 23" ) O's Az. 

" " 27 48 42 J N.88°E.; 

log 8.8239 
log 3.4622 
ar. co. log 7.4137 
1. cosec 0.0003 
1. cos 9.7001 

If A (ti—h) = 10", A log (h f -h) = A 1. cos L = .0015, and 
JX = 6'. If the difference of altitudes can be depended on 
within 5", the latitude is correct within 3'. 

2. 1865, July 13, 5 h P. M., in lat. 54° 20' N., long. 113° W., 
by account ; the altitude of the sun's lower limb was ob- 
served at h 23 m 34 s by the chronometer, which was slow of 
G. mean time 10 m 18 s ; and the sextant remaining clamped 
the upper limb arrived at the same altitude at h 27 m 8 s -5 ; 
the true altitude of both limbs was 27° 18' 20"; required 
the latitude. 

The sun's diameter, 31' 33", is the difference of altitudes 
in this case. The sun's azimuth computed from the altitude 
and supposed latitude is 1ST. 88|° W. 



required 


; 11 19 51 .0, 
the latitude. 




h'—h = 
t — 
z- 

L = 


i 

48' 19" 
6 m 2o 8 .7 
88° 
59° 55' N. 



LATITUDE BY TWO ALTITUDES. 257 



TT 



log 8.8239 

h — h' = W 33" log 3.2^2 

t = 3 m 34 s .5 ar. co. log 7.6686 

Z— 88£° 1. cosec 0.0002 

L = 53° 56' N. 1. cos 9.7699 

If we suppose t to be in error I s , 1. cos L will be in error 
.0020 and Z, 11'. If the elapsed time can be depended on 
within s . 5, the latitude is correct within 6'. 

The longitude obtained from the same observations is 
113° 5' W. 

This method of observing the successive contacts of the 
two limbs of the sun with the horizon with the sextant 
clamped is recommended. 

254. Problem 60. To find the latitude from two altitudes 
of different bodies, or of the same body when the change 
of declination is considerable, the Greemoieh times being 
known. 

Solution. The observed altitudes should be reduced 
to true altitudes, watch times to chronometer times, and the 
difference of the two chronometer times for the rate in the 
interval, as in Problem 59 (Art. 238), and to a sidereal in- 
terval, when the altitudes of two different bodies have been 
observed. 

When the latitude only is to be found, the Greenwich 
mean times of the observations are wanted only with suffi- 
cient exactness for finding the right ascensions and declina- 
tions of the bodies. If the longitude is also to be found, it 
is necessary to note the times by a Greenwich chronometer, 
or a watch compared with it. 

It is well also to note the azimuth, or bearing, for each 
observation ; or, as is sufficient, the difference of the azi- 
muths. 

Let M and M' be two positions of the body, or bodies, 



258 



NAVIGATION. 




h — 90°— Z M, the true altitude 
of M, 

h! = 90°— Z M', the true altitude 
of M', 

d = 90°— P M, the decimation 
of M, 

d'=90°— P M', the declination 
of M', 

r=ZPM, the hour-angle of M, 

f = ZP M', " « ' " M ; , 

t == T-T= M P M', the differ- 
ence of the hour-angles. 

£ is positive in the direction of the diurnal rotation, and 
will be positive and less than 12 h if M P M', estimated from 
P M in that direction, is less than 12 h , or 180°. We shall, 
as in the diagram, designate the two positions as M and M' 
respectively, so as to satisfy that condition. It will be seen 
hereafter, however, that it will be sufficient to have t nu- 
merically less than 12 h , without regard to its sign. 

255. The method of finding t varies with the objects ob- 
served. But in any case we are at liberty to add or to sub- 
tract 24 h , either to change a negative into a positive result, 
or to reduce it within the numerical limit of 12 h . A posi- 
tive result greater than 1 2 h , or a negative result less than 
12 h indicates that PM' is in the negative direction from 
P M. 

a. When two bodies are observed at different times, if 

a and a' are their right ascensions, 

S and /S", the sidereal times of the observations, 

by Art. Ill, 

T=S -a, T' = 8'-a f 

and t = S'—S—a' + a; 

or, t = s-ha— a! (208) 



LATITUDE BY TWO ALTITUDES. 259 

when M has been first observed ; 

t= — (s + a'—a) (208') 

when M' has been first observed. 

In either case, the right ascension of the body first ob- 
served is added to the sidereal interval, and the right ascen- 
sion of the other body subtracted. If M' has been observed 
first, the sign is to be changed. 

b. When two bodies are observed at the same time. 

5 = and t = a—a\ (209) 

the difference of their right ascensions. 

c. When the sun, moon, or a planet is observed at two 
different times, we have, as in Art. 238, 

t — s—s. 4 a (1 — .00273) (210) 

for an elapsed sidereal time ; 

t = t m + t m (9S.8565- J h a) (211) 

for an elapsed mean time; in which A h a is the change of 
right ascension in l h of mean time; (1 — .00273) A h a, the 
change of right ascension in l h of sidereal time ; and t m and s 
as coefficients are expressed in hours. 

In the case of the sun the last expression becomes, as in 
(175) 

t=t m +t m . 4E; 

in which J h E is the hourly change of the equation of time, 
employing for E the sign of its application to mean time. 

If this result exceed 12 h , it should be subtracted from 
24 h . M' in that case is the position at the first observation. 

256. We have given (Fig. 47) 

h = 90°-Z M, xl = 90°-P M, 

h' = 90°-Z M ; d l - 90°-P M', 

and ^MP M', 



260 



NAVIGATION. 



to find from the triangles P M M', Z M M' and PM'Z, or 
PMZ, 

L = 90°-P Z. 

The following method is selected as the most common. 
25 7. Fourth Method of Bote ditches Navigator. 
1. In the triangle P M M', (Fig. 48) 

MPM'-«, P M = 90°- J, PI' = 90°-c?', 

are given, from which w T e may find 

M M' — _5, the distance of the two positions, and 
the angle, PM'M = P, 

By Sph. Trig. (4) and (10)* we have 

cos B = sin d! sin c?+cos d! cos d cos t, 
cos dJ tan d — sin d' cos t 



cot F : 



sin t 



the 2d of which, by multiplying both numerator and deno- 
minator by cos d, becomes 

™ cos d f sin d — sin d' cos d cos t 
cot P f — v- v— . 

cos <x sin £ 

To adapt these to logarithmic computation, put 

rn sin Jtf ' == sin d ) 

m cos M ' == cos c? cos £ f 

and we shall have, after eliminating m, 

tan M' = tan c? sec £ 

cos 

cot * sin (if — £') 



^ sin <# cos (M ' — d') 

sin ilf 



COt F : 



cos if 7 



\ ( 212 ) 



* Cos a = cos b co3 c + sin b sin c cos -4, (4) 

sin A cot B = sin <? cot b — cos c cos A (10) 



LATITUDE BY TWO ALTITUDES. 261 

We may take M' in the same quadrant as t and give to it 
the sign of d. - — jp is then positive, and B will be in the 
1st or 2d quadrant as the numerical value of M'—d'. 
jp is also positive, and JP r will be in the 1st or 2d quad- 
rants according as M'—d 1 has the positive or negative sign. 

If M be drawn perpendicular to P M', (Fig. 47) we shall find 
90° — P = J/ 7 , the declination of 0, and M' = If— d' . 

2. In the triangle Z M M', 

ZM= 90°- h, Z M'= 90°- h\ M W=B, 
being known, if we put Z M' M = Q\ we have, by Sph. 
Trig.* (30) 

/sin i(90° -h + 9Q°— fc'+ B) sin i(90° — &— 90°+ h'— B)\ 



sini£'=|/(- 



cos h' sin B 

and by reducing 

. in ,_ /(cos i(B+h' + h) sin i(B+h f —h) 

sin 2 u -y \- - cos h , sin B 

or, putting 



sm 



s = i (JS + h' + h), ) 

/cos s sin (5 — fc)\ # [ (213) 



i€'=|/(- 



cos A sin B J ' ) 
Since this radical may have either the positive or nega- 
tive sign, we may take | Q' in either the 1st or 4th quad- 
rants, or numerically less than 90° with either sign. We 
shall thus have two equal values of Q' with opposite signs. 

3. PM'Z = PM'M-ZM'M (Fig. 47), or representing 
it by q\ 

for which we shall have two values, resulting from the two 
values of Q', which are indicated by 

q' = F^Q'. (214) 

* sh f i A = Bini(«+5- e )smi(a-6+ e ) 

sin o sin c v ' 



262 NAVIGATION. 

In the triangle P M' Z, 

P M'= 90°- d', Z M' = 90°- h\ PM'Zz: q\ 

being known, we have, by Sph. Trig. (4), for finding 

Z=90°-PZ, 
sin L = sin h! sin d' + cos h! cos d' cos q\ 

To adapt this for logarithms, put 

n' sin iV 7 — cos h' cos q' ) 
n' cos i\T'= sin h\ ) 

and we shall have, after eliminating n\ 

tan JV r = cot A' cos q' ) 

. 7 -__ sinA / sin(i^ / + ^) >• (215) 

sin jLj — ■ ^77 • i 

cos N ! ) 

We may take JV numerically less than 90°, and give it 
the same sign as that of cos q r : and the latitude i, numeri- 
cally less than 90°, with the same sign as J¥' + d r . 

There Avill be two values of L derived from the two values 
of q'. Unless q f is small, we may select the value w r hich 
agrees best with the known approximate latitude. 

If Z n (Fig. 47) be drawn perpendicular to P M', we shall find M' n =N' 
and 90° — Y n =N'+ d\ the declination of n. 

258. Thus, by (212), (213), (214), and (215), the solution 
is effected. We have seen in each how the proper quadrant 
of the unknown quantities can be 'determined (with the re- 
striction of t to positive values less than 12 h ), except that Q' 
may have two values. The same results would be obtained 
by following the usual trigonometric precepts. 

259. We may, however, select the proper value of Q' and 
avoid the double solution, by means of the noted azimuths. 
For, if Z and Z are the azimuths of M and M' (Fig. 47), 
reckoned as positive toward the right, 

MZM'=Z'-Z, 
and in the triangle M Z M' 



LATITUDE BY TWO ALTITUDES. 



263 



sin Q'=$in (Z'-Z)~. 

As cos h and sin _Z? are positive, (?' will have the same sign 
as (Z'—Z) restricted numerically to 180°. Hence, as Q' is 
to be subtracted from P\ Ave shall have 

q'=_P f — Q\ when M' bears to the right of M, 
q'=P r + Q', when M' bears to the left of M. 
Figs. 47 and 48 illustrate these two cases, for, in the first, 





where M' is to the right of M, PM'Z = P M' M - Z M' M ; 

and in the second, where JVT is to the left of M, 
PM'Z=PM'M-ZM'M. 

If M M' be extended to the meridian, in the first case, 
P and Z are on the same side of the intersection, and in 
the other they are on opposite sides ; so that 
q f =jP f — Q' when the zenith and north pole are on the same 

side of the great circle, which joins the two 

positions ; 
q r =P'+ Q\ when the zenith and north pole are on different 

sides, or the zenith and south pole are on the 

same side, of that circle. 
To use this criterion it will be necessary to note where 
the circle, which connects the two positions observed, crosses 
the meridian. 



264 NAVIGATION. 

The doubtful case with either of these criterions is when 
Z'—Z is near 0, or 180°, or the great circle, which joins the 
two positions, passes near the zenith. These two conditions 
are coincident, except when the two positions are near the 
meridian on the same side of the zenith. 

260. The hour-angle of M' may also be found, and thence 
the longitude, if the times have been noted by a chronometer 
regulated to Greenwich time. For, in the triangle P M' Z, 
we have, by Sph. Trig. (10), 

, rrt , cos d' tan Ji r — sin d' cos a' 

COt T = : ; — — , 

sin q 
which, by multiplying numerator and denominater by cos h y 

becomes 

, rril cos d' sin h' — sin d' cos h r cos q f 

cot T — tt~ -, — - 

cos h sin q 

Putting, as before, 

n' sin JSF= cos h' cos q ! 

?i f cos iV'= sin h\ 
and eliminating n\ we have 

tan i\r= cot h! cos q' ) 

cot r= '<*£ cos gT ±*1. \ 

sin Is ) 

Or, L having been found, we have also 

. m1 sin q' cos h f /^h\ 

sm T — — - — ^ — . (217) 

cos L x 7 

By (217) sin T and sin q' have the same sign, which will 
be positive when M' is west of the meridian, negative when 

M r is east of the meridian. In (216), if JST r has been taken 

cos q' 
less than 90° with the same sign as that of cos q\ — — ^7 is 

positive, and cos T' and cos (iV'-f- d!) have the same sign. 
We may take T\ then, numerically in the same quadrant as 
iV r/ + d\ and give it the positive sign when q 1 '< 180°, the 
negative sign when q' > 180°, or is negative. 

If the proper value of Q ', and therefore q\ has not been 



(216) 



LATITUDE BY TWO ALTITUDES. 



285 



previously determined, we shall have two values of T\ but 
may ordinarily take that which agrees best with its known 
approximate value. 

261. The preceding formulas employ the angles at M', and 
the triangle P M' Z. We may also use the angles at M and 
trie triangle PMZ; and, as M and M' are similarly situated 
with regard to the triangles, except that the angles at each 
are estimated in opposite directions, Ave shall obtain, by 
interchanging accented and unaccented letters in the pre- 
ceding formulas, a set similar in form, but with this differ- 
ence of interpretation, that t is posi- 
tive in the opposite direction of the 
diurnal rotation, and q is less than 
180° east of the meridian and great- 
er than 180°, or negative, west of 
the meridian. This difference is 
shown in Fig. 49, in which the 
primitive position of the triangles 
is east of the meridian, instead of 
west as in Fig. 47. 

We have, then, 

tan M~ tan d' sec £, 

sin d' cos (M — d) 




cos B 
cot P = 



sin M 
cot t sin (M- 



•d) 



Bin f 



cos M 
fcos s sin (s — h') 



hQ = i/(- 



cos li' sin B 
tan iV= cot h cos q, ) 



sin L : 



cot T-- 



sin li sin (iV+ d) > 
' coTiV^ ' ) 

cot q cos (JV+ d) 
sin JV" 



(212') 

(213') 
(214') 
(215') 

(216') 



266 



NAVIGATION. 



sin T = — 



sin q cos h 



(217') 



q — P—Q, when M bears to the left of M', 
q = P + Q, when M bears to the right of M', 

262. Either set of formulas may be used ; but, in general, 
the latitude can be best found from the altitude, which is 
nearest the meridian ; the hour-angle, from the altitude 
which is nearest the prime vertical. 

The distinction made with regard to M and M' (Art. 254), 
is important only so far as it may aid in determining the 
hour-angles and selecting the proper value of q or q'. So 
that it is sufficient practically to find t numerically less than 
12 h without regard to its sign. 

263. The most favorable condition is, as stated in Article 
(243), when the difference of azimuths is 90°. But altitudes 
near the meridian will give a good determination of the lati- 
tude, and altitudes near the prime vertical, a good deter- 
mination of the hour-angles, when the difference of azimuths 
is small, or near 180°; especially if the altitudes have been 
carefully observed, and their difference is nearly exact. 

264. If we put in (212' &c.) 
M=-A, P = C, 



Q=Z, 

we shall have 



2=:90°-#, 

tan A = — tan d' sec £, 

~ sin d' cos (A + d) 

cos G — 



P=90°-P, 



cot F- — 



sin A 
tan t cos A 



sin | Z = j/7- 



sin {A + dy 
/cos s sin (s - 



-A') 



cos h' sin B 
G = F±Z, 
tan 1= cot h sin 6r, 

T _ sin h sin (d + I) 

-Lj — — 



SID 



COS / 



(218) 



LATITUDE BY TWO ALTITUDES. 267 

the formulas of Bowditch's 4th method, if we take h r and^c?' 
as the greatest altitude and the corresponding declination. 

By attending to the signs of the quantities and their func- 
tions, the proper result can be obtained. We may give to 
Z the same sign, or name, as that of the latitude, when the 
zenith and elevated pole are on the same side of the great 
circle, which passes through the two j30sitions observed ; 
but a different sign, or name, from that of the latitude, when 
the zenith and elevated pole are on opposite sides of that 
circle. 

The precepts, which Bowditch gives (p. 194) are based 
on the consideration of the trigonometric functions, and 
possess the advantage that the sum of quantities of the same 
name, or the difference of quantities of different names, is 
taken, and the name of the greater given to the result. If 
the sum exceed 180°, it should be subtracted from 360°, and 
the name changed. 

Example. 

At sea, 1865, May 5, 7 h P. M., in lat. 36° 41' N"., long. 
168° 57' W., by account; altitudes of a TJrsse Majoris and 
the moon were observed and the means noted as follows : 
required the latitude and longitude. 

T. by Chro.6 h 41 m 27 s : alt, of* 62° 18'30 r/ ; bearing N".byE.|E. 
" " " 6 53 8 ; " " JD44 56 50 ; " S.E.fS." 
Chro. at 18 h fast of G. m. t, 36 m 48 s , losing daily 10 s .2 ; 
Index cor. of sextant —4' 20"; height of eye 18 feet. 
Ship running NT. E. by X. (true), 10 knots an hour. 

From these data, reducing the second altitude to the posi- 
tion of the first, we find, 

a Ursce Maj. Moon, 

G. ra. t. May 5 18 h 4 m 39 3 .3 IS 11 16 m 20 s .4 Elap. in. t. ll m 41U 

R. A. 10 55 24.3 11 37 57.0 Red. +1.9 

Dec. d'— + 62° 28' 45" d=—0° 53' 54" -Elap. sid t.-ll 43 .0 
True alt. h'= 62 9 29 h= 45 39 54 Diff. ofR. A. 42 32.7 

*=30 49.7 



268 


NAVIGATION. 




Computation by (212- 


-216). 


t- 7° 42' 26" 


1. sec t 


0.00394 




d— — 53 54 


1. tan d 


8.19535 71 


1. sine* 8.19530 n 


M' = — 54 24 


1. tan M-' 


8.1992971 


1. cosec 31' 1.80077 n 


d'=+ 62 28 45 








M-d' = - 63 23 9 






lcos(M'-d') 9.65126 


.5=: 63 38 39 


1. cosec B 


0.04766 


1. cos B 9.64733 


h= 45 39 54 






1. cot t 0.86859 


h'= 62 9 29 


1. sec 7i' 


0.33066 


1. sec M 1 0.00005 


2s= 171 28 2 






Ism(M'-d') 9.95136 n 


s= 85 44 1 


1. cos s 


8.87153 


1. cotP' 0.82000 n 


*—A = 40 4 7 


1. sin (s—h) 


9.80869 
19.05854 




i Q = 19 46 19 


1. sin -J- Q' 


9.52927 




G'= 39 32 38 








P' = 171 23 36 


1. cot A' 


9.72278 




q'— 210 56 14 


1. cos q' 


9.93335 ^ 


1. cot q' 0.22230 


or -149 3 46 


1. tan iV' 
1. sin ti 


9.65613 n 


1. cosec N' 0.38441 n 


JST'z=— 24 22 20 


9.94657 


1. cos (N' + d') 9.89590 


d'= + 62 28 45 


1. sec iV" 


0.04054 


1. cot T' 0.50261?i 


N' + d'—+ 38 6 25 


1. sin (N'+d') 9.79038 


r'=r — l h 9 m 48 8 




1. sin L 


9.77749 


■X-'sR.A.= 10 55 24.3 
Sid. time 9 45 36 .3 


Lat. 36 c 


' 48'.3 N. 




— S —2 53 28.5 


Long. 168 


52 .4 W. 




Red.forG.m.t. — 2 58.2 
L. m. t. 6 49 9 .6 
G. m. t. 18 4 39 .3 
Long. 11 15 29.7 



265. When equal altitudes have been observed, (213) and 
(213') reduce to the simple form, 

// CQg-ftJ?+»)\ , 

y 1 2 ^ i » ™° *' 



sin £ (?'= sin | # : 



cos £ P cos hj ' 

We have also from the isosceles triangle Z M M' (Fig. 47), 
cos Q'= cos Q = tan J- ^ tan A. (220) 

266. When a lunar distance has been measured and re- 
duced to the true or geocentric distance (Prob. 55), we have 
in the triangle P M M' (Fig. 47), 



LATITUDE BY THBEE ALTITUDES. 269 

PM = 90°-e?, 3PM'=-90°— : dr," and 

JM M'=j5, the geocentric distance, 
from which we may find P M' M = P' and P M M'= P. 
By Sph. Trig* (30), 

sm ^ -y [ sin b cos d' : ) 

which reduces to 

• i™, //cos j(B+d+d') sin j(B+d '-d)\ 
Sin ^r - y ^ sin i? cos # J 

or putting 



(221) 



"»*'=•( -•*££?' ) l 

So also we shall have 

• i r> //cos s' sin ($' — d')\ #««,>.% 

m *. Pas .r \ ^ ^ Id > < 221 ) 

These may be employed instead of (212) and (212') ; and 
if the altitudes of both bodies have been observed, the lati- 
tude and hour-angles can be found by the subsequent formu- 
las. 

From a lunar distance, then, and the two observed alti- 
tudes, the longitude, latitude, and local time may all be 
found. 

267. PKOBLEMf 61. To find the latitude from three alti- 
tudes of the same body near the meridian, and the chrono- 
meter times of the observations. 

The Greenwich time, or the longitude, will be required 
only with sufficient exactness for taking the declination from 
the Ephemeris. 



* smH A = ^*(«+»7«)Bh *(«-»+«) (80) 

sin o sin c v ' 

f Chauvenet's Astronomy, Vol. L, p. 299. 



270 



NAVIGATION. 



Let A, h\ h\ be the true altitudes, 
h 0J the meridian altitude, 
Tj T\ T", the chronometer times, 

T^ the chronometer time of meridian passage, 
a, the change of altitude in I s of chronometer 
time from the meridian, 
then we have, from the three observations (120), the differ- 
ences of the times being expressed in seconds, 

h Q = h +a(T-T )*) 

h = h'+a(r-T o y\ (222) 

V=A f +a(Z*-JS) , J 

the differences of which are 

= h> -7i + a [(r -T o y-(T -T yi 
= £"-# + a [(T f '~Ty-(i«^To)% 

From these we obtain 
7i — 7i 



T- 
h'- 



-T 



a (T'+T)-2a T oy 



(223) 



(224) 



TF — f7 = a(T»+T')-2aT , 

the difference of which is 

y_y h-h' _, Tlf T , 

rpn rpi rpi rp t(/ yJ. J. ]• 

If now we put 
h y 

b = T , T , the mean change of altitude in I s of the chro- 
nometer from the first to the second observa- 
tion, 

c = rpn T n tne mean change in I s from the second to the 

third observation, 
we shall have, from (224) and (223), 
_ c—b 



2 o/ 



(225) 



LATITUDE BY THREE ALTITUDES. 271 

from which a and T Q may be fonnd. h may then be found 
by either of the three equations (222) ; and thence the lati- 
tude as from an observed meridian altitude (Prob. 45). 

The correction and rate of the chronometer need not be 
known ; it is sufficient to have the rate uniform during the 
period of the observations. 

This method is restricted like other methods of circum- 
meridian altitudes (Arts. 150, 247), to hour-angles varying 
with the meridian zenith distance of the body. Its accuracy 
depends upon the precision with which a, &, and c are ob- 
tained : hence the altitudes should be carefully observed, so 
that their differences shall be nearly exact, and the intervals 
of time should be greater, the greater the distance of the 
middle observation from the meridian. 

268. The computation is facilitated if the observations are 
made at intervals of exact minutes of time. For then, ex- 
pressing these intervals in minutes, and taking a, 5, and c as 
changes of altitude in l m of chronometer time, 

, 7i — ~h! ~h! — h" ^ _ c — h , , 

h c 

are easily found. - — and - — , however, will be in minutes 

Z co l co 

of time. Reducing them to seconds, we shall have, instead 
of (225), 

CO CO 

In (222) 

h = h +a(T-T o y\ 
h = h' + a(F-T o yy 

h Q = h n +a(T"-T Q y) 
we may now use, in computing the reductions, a table of 

* If the intervals are reduced by the methods of Art. 238 to intervals of 
hour-angle, and the declination has not changed, a will be the change of alti- 
tude in l m of hour-angle, as in (119) and Tab. XXXII. (Bawd.). 



272 NAVIGATION. 

" squares of minutes and parts of a minute" as Tab. XXXIII. 
(Bowel.). 

269. For the sun, as its declination usually changes in the 
interval, T is the chronometer time of the maximum alti- 
tude* (Art. 142) ; in which case the meridian declination is 
to be employed. 

If the latitude and longitude also have changed, as is usu- 
ally the case at sea, c and 5, being observed changes of alti- 
tude, are no longer due to the diurnal rotation alone, but are 
affected by the change of position. But with altitudes near 
the meridian, a change of latitude has the same effect as a 
change of declination in the opposite direction, while a 
change of longitude is equivalent to a change in the rate of 
the chronometer. If, then, the motion of the ship has been 
tolerably uniform in the interval of the observations, T is 
still the chronometer time of the maximum altitude. The 
method, then, can be used at sea when the sea is smooth 
and the horizon well defined, and meridian altitudes of the 
sun are prevented by passing clouds. But the altitudes 
should be very carefully observed, and on both sides of the 
meridians when practicable. The intervals should not be 
less than 10 m . 

270. If the three altitudes are observed at equal intervals 
of time, the process of computation becomes much simpli- 
fied.! 

Let t be this common interval, 

T 7 , the time from the maximum altitude at which the 
second observation w r as made ; 
then we have 

h -h + a (T-ty 

h = 7i f +a T 2 

h = h f '+ a (T+t) 9 . 



* Chauvenet's Astronomy, Vol. I., pp. 299 and 244. 
f Chauvenet's Astronomy, Vol. I., p. 309. 



LATITUDE BY THREE ALTITUDES. 273 

Half the sum of the second and third equations is 

which, subtracted from the third, gives 

= h f - i (h + h") - a f; 
whence 

a f = h'- \ (h + h°). 

The difference of the first and third gives 

al - T? ' 
which, substituted in the second equation, gives h . 

If we put A— a f, we have, as the formulas for computa- 
tion, 

EXAMPLES. 

1. At sea, 1865, Sept. 16, in lat. 40° 0' K, long. 60° 0' W., 
by account ; the following altitudes of the sun were observed 
near noon ; index cor. + 2' 10*; height of eye 20 feet. 

T. by Chro. 4 h 25 m 15 9 Biff. Obs'd alt. of O. 52° 24' 20" (S.) Biff. 



32 

37 


15 
15 


5 






20 10 
14 50 


4' 10' 
5 20 


T+ T') = 4* 28 re 


l 45 s 




. 250" 


= S5".1 


log6 


1.553 


^^1= -7 

a 


34 




_ 320" 
5 


= 64 .0 


log 30 


1A11 


T Q = 4: 21 


11 




_ 28".3 
~12~ 


= 2 .36 


ar. co. log 


a 9.627 


T—4: 25 


15 






454 s 


, 30 6 
log— — 
a 


2.657 


T-T Q = 4 


4 




a(T-T Q f 


= 2".36 x 


16.5 = 39" 





274 NAVIGATION. 

1st alt. of 52° 24' 20" ( In. cor. +2' 10" dip —4' 24" 

+*13 45 -J S. diam. +15 58 ref. & par. —38 
Mer. alt. h = 62 38 5 (Red. +39 

Mer. zen. dist. z = 37 21 55 N. 
Mer. dec. d — 2 28 48 N. 

Lat. 39 50 43 N. 

If the middle altitude had been 30" greater or less than 52° 20U0", the 
result would have been varied only 20" ; but if the middle altitude had been 
1/ less, the latitude would have been 39° 40'. The interval is too short, un- 
less the differences of the altitudes can be relied on within 40". 

2. At sea, 1865, May 8, in lat. 35° 50' S., long. 60° 0' E., 
by account ; the following altitudes of the sun were observed 
at equal intervals near noon : index cor. + 2' 0* ; height of 



eye 


20 feet. 






















Obs'd alt. 


of 0.36° 


44' 


20" (N.) 


T. by watch ll b 50 m 


20 s 








36 


51 


40 






12 





20 








36 


52 


40 






12 


10 


20 




*(* + 


70 = 


:36 


48 


30 


h- 


-h" = 


-8' 20" 










A = 




3 


10 


i (*-; 


h")=. 


- 125" 


2 log 4.194 
log A 2.279 






h' = 


:36° 


51' 


40" t 


Red. 


+ 1' 


22" 




1.915 








+ 13 


42 -J 


In. cor. 


+ 2 


Dip 




-4' 24" 




Mer. alt. 


h = 


:37 


5 


22 ( 


S. diam 


. + 15 


53 Ref. & 


par.- 


-1 9 




Mer. zen. dist. 


h — 


52 


54 


38 S. 














Mer. dec. 


d = 


■11 


1 


19 N 
















Lat 35 


41 


19 S. 













If either of the altitudes be changed 1', the reduction to the meridian will 
be changed less than 40" : so that, if the differences can be depended on 
within 1', the reduction is correct within 40". 



CHAPTER X. 

AZIMUTH OF A TERRESTRIAL OBJECT. 

271. Ix conducting a trigonometric survey, it is necessary 
to find the azimuth, or true bearing, of one or more of its 
lines, or of one station from another. Thence, by means of 
the measured horizontal angles, the azimuths of other lines 
or stations can be found ; and, still further, a meridian line 
can be marked out upon the ground, or drawn upon the 
chart. 

For example, suppose at a station, A, the angles reckoned 
to the right are 

B to <7, 48° 15' 35"; C to D, 73° 37' 16"; D to E, 59° 45' 20"; 
and that the azimuth of D is N". 35° 16' 15" E. ; the azimuths 
of the several lines are 

A B, N\ 86° 36' 36" W. A Z>, N. 35° 16' 15" E. 

A (7, N. 38 21 1 W. A E, N". 95 1 35 E. 

If upon the chart a line be drawn, making with A B an 
angle of 86° 36' 36" to the right, or with iDan angle of 
35° 16' 15" to the left, it will be a meridian line. 

Or, if a theodolite or compass be placed at A in the field, 
and its line of sight, through the telescope or sight-vanes, be 
directed to D, and the readings noted ; and then the line 
of sight be revolved to the left until the readings differ 
35° 16' 15" from those noted,, it will be directed north. If a 
stake or mark be placed in that direction, it will be a meri- 
dian mark north from A. 



276 



NAVIGATION. 



212. If the azimuth of a terrestrial object is known, it 
may be conveniently used in finding the magnetic declina- 
tion, or variation of the compass. For, let the bearing of 
the object be observed with the compass, — the difference of 
this magnetic bearing and the true bearing is the magnetic 
declination, or variation, required. It is east if the true 
bearing is to the right of the magnetic bearing ; but west if 
the true bearing is to the left of the magnetic bearing.* 

273. Peoblem 62. To find the azimuth, or true bearing, 
of a terrestrial object. 

Solution. Let 
Z (Fig. 50) be the zenith, or place, of 

the observer ; 
O, the terrestrial object ; 
M, the apparent place of the sun, or 

some other celestial body ; 
Z — N Z O, the azimuth of O ; 
z = N Z M, the azimuth of M ; 
^ v = Z-s = MZO, the azimuth an- 
gle between the two objects, or 
the difi&rence of azimuth of M 
and O. 
The problem requires that z and £ be found ; then we have 

Z=z + $. 
Or, numerically, 




* This has reference to the two readings. The actual direction of the 
object is the same ; but the true and magnetic meridians, from which the 
angles are estimated, are different. When the magnetic declination is east, 
the magnetic meridian is to the right of the true meridian ; when the mag- 
netic declination is tvest, the magnetic meridian is to the left of the true 
meridian. 

It is sometimes necessary to distinguish between the magnetic bearing and 
the compass bearing. The latter is affected by the errors of the instrument 
employed and by local disturbances ; the former is free from them. 



AZIMUTH OF A TERRESTRIAL OBJECT. 277 

Z=2 + £, when the azimuth of the terrestrial object is 

greater than that of the celestial, 
Z — z — ^ when it is less. The sign of £ should be noted 

in the observations. 

274. 2 = XZ M, the azimuth of the celestial body, may- 
be found from an observed altitude (Prob. 40), or from the 
local time (Prob. 38). In the first case, the most favorable 
position is on or nearest the prime vertical ; for then the 
azimuth changes most slowly with the altitude. In the 
latter, positions near the meridian may also be successfully 
used. 

275. £ = M Z 0, the azimuth angle between the two ob- 
jects, may be found in one of the following ways : — 

1st Method. (By direct measurement.) 

M Z O, being a horizontal angle, may be measured direct- 
ly by a theodolite or a compass, by directing the line of sight 
of the instrument first to one of the objects and reading the 
horizontal circle, then to the other and reading again. The 
difference of the two readings is the angle required. Or, the 
telescope or sight-vanes of a plane table may be directed 
successively to the objects, and lines drawn upon the paper 
along the edge of the ruler in its two positions, and the an- 
gle which they form measured by a protracter. 

At the instant when the observation is made of the celes- 
tial object, either its altitude should be measured, or the 
time noted, so as to find its azimuth simultaneously. 

The instrument should be carefully adjusted and levelled. 
With the compass or plane table, it is not well to observe 
objects whose altitudes are greater than 15°. 

A theodolite can be used with greater precision than the 
other instruments ; but the greater the altitude of the object, 
the more carefully must the cross-threads be adjusted to the 
axis of collimation, and the telescope be directed to the 
object. 




278 NAVIGATION. 

The error of collimation is eliminated by making two ob- 
servations with the telescope reversed either in is Vs, or by 
rotation on its axis. Low altitudes are generally best. 

276. If the sun is used, each limb may be observed alter- 
nately ; or a separate set of observations may be made for 
each. 

To find the azimuth reduction for semi-diameter, when 
but one limb is observed ; 

Let h— 90° — Z s (Fig. 51), the altitude of 

the sun, 
s = S s, its semi-diameter, 
s f = S Z s, the reduction of the azimuth 

for the semi-diameter. 

We have 

. sin S s 

sin- SZs = .., „ 3 

sin Zs 

or, since s and s r are small, rig - 51 - 

s f — s sec A, (229) 

which is the reduction required. 

The sign, with which it is to be applied, depends upon 
the limb observed. 

277. If the observations are made at night, and the ter- 
restrial object is invisible, a temporary station in a conve- 
nient position may be used, and its azimuth found. The 
horizontal angle between this and the terrestrial object may 
be measured by daylight, and added to, or subtracted from, 
this azimuth. 

A board, with a vertical slit and a light behind it, forms 
a convenient mark for night observations. 

The place of the theodolite should be marked, that the 
instrument may be replaced in the same position. But in 
doing this, ancl selecting the temporary station, it should be 
kept in mind that a change of the position of the instru- 



AZIMUTH OF A TERRESTRIAL OBJECT. 



279 



ment of g^Vs °f tne distance of the object may change the 
azimuth 1'; or of 2W000 of the distance may change the 
azimuth more than l" . 

278. 2d Method. Finding the difference of azimuths of a 
celestial and a terrestrial object by a sextant y sometimes 
called an " astronomical bearing" 

Measure with a sextant the angular distance M O (Fig. 52) 
of the two objects, and either note the time by a watch 
regulated to local time, or measure simultaneously the alti- 
tude of the celestial object. Measure, also, the altitude of 
the terrestrial object (if it is not in the horizon), either with 
a theodolite which is furnished with a vertical circle, or 
w T ith a sextant above the water-line at the base of the ob- 
ject, when there is one. Correct the readings of the instru- 
ments for index errors, and when only one limb of the sun 
is observed, for semidiameter.* 

Observed altitudes of either object above the water-line 
are also to be corrected for the dip by (53) or Tab. XIII. 
(Bowd.), if the horizon is free; but by (55) or Tab. XVI. 
(Bowd.), if the horizon is obstructed. 
The altitude of the celestial object, 
when not observed simultaneously, 
may be interpolated from altitudes 
before and after, by means of the 
noted times. (Bowd, p. 246.) Or 
the true altitude may be computed 
for the local time (Prob. 38 or 39), 
and the refraction added and the 
parallax subtracted to obtain the 
apparent altitude. 




* It is best in measuring the distance of the sun from the terrestrial ob- 
ject to use each limb alternately. 



280 NAVIGATION. 

Let h! = 90° — Z O (Fig. 50), the apparent altitude of O, 
H' — 90° — Z M, the apparent* altitude of M. 
D — M O, the corrected distance. 

We have then in the triangle M Z O, the three sides 
from which £ = M Z O, may be found by one of the follow- 
ing formulas : — 

1. By Sph. Trig. (164) we have 

. 1V /sin i (Z> + 7T— h') sin HZ) — H' + h') 
sm | £ = j/- cqs ^ cqs v 1 



or, letting d == H'—h\ 
sin 



i >• - / sin Hi) + ^) sin j (2> — <Z) 
2 s ~~ T ~ cos H' cos # 



(230) 



2. By Sph. Trig. (165). 

. y ± / cos \{H' + h' + J9) cos i (H' + h' — D) 
cos 45 - j/ c~os iT cos A' 

or, putting 

s = H^' + # + i?) 1 

, v / cos s cos (s — Z>) 1* (231) 

COS J £ = i/ ^7^ TT 1 | 

r COS jH cos h' J 

(230) is preferable when £ < 90°; (231), when $ > 90°. 

279. If O is in the true horizon, or its measured altitude 
above the water line equals the dip, h = 0, and the right 
triangle M m O gives 

cos $ = cos MO = cos D sec H' ; (232) 

or more accurately when £ is small (Sph. Trig., 105), 

tan \ $ = V (tan J {D + H f ) tan £ (D-H f ) ). (233) 

If the terrestrial object is in the water-line, A' is negative, 
and equals the dip. 

* The true altitude of M is used in finding g, its azimuth. 



AZIMUTH OF A TERRESTRIAL OBJECT. 281 

280. If both objects are in the horizon, or H and h are 
equal and very small, we have simply 

$ = Z>. (234) 

In general the result is more reliable the smaller the in- 
clination of M O to the horizon. If M O is perpendicular 
to the horizon, the problem is indeterminate by this method. 

281. If the terrestrial object presents a vertical line to 
which the sun's disk is made tangent, the reduction of the 
observed distance for semidiameter is 

s' = s sin MOZ (235) 

and not s, the semidiameter itself. This follows from the 
sun's diameter through the point of contact, O, being per- 
pendicular to the vertical circle Z O and not in the direc- 
tion of the distance O M. 

As the altitude of the terrestrial object is always very 
small, we may find MOZ by the formula 

^ r ^ „ sin h' 
COS MOZ— - — — ; , 

sin D 7 

D' being the unreduced distance. 

282. When precision is requisite, the axis of the sextant 
with which the angular distance is measured must be placed 
at the station Z ; and if the object seen direct is sufficiently 
near, the parallactic correction must be added to the sex- 
tant reading. If 

J represent the distance of the object, 

cZ, the distance of the axis from the line of sight or axis of 
the telescope, this correction is 

p = — cosec V = 206265" — . (236) 

It is 1', when A = 3437.75 cl 

283. If the distance of the terrestrial object and the dif- 
ference of level above or below the level of the instrument 



282 NAVIGATION. 

are known, Ave may find its angle of elevation, nearly, by 
the formula 

tan h' = — , 
A 

A being the distance of the object, and 

E, its elevation above the horizontal plane of the instru- 
ment. 
If the object is below that plane, E and h! will have the 

negative sign. 

Note. — The horizontal angle between two terrestrial objects may also be 
found by measuring their angular distance with a sextant, and employing 
the same formulas (230 to 234) as for a celestial and terrestrial object ; 
H' and K representing their apparent angles of elevation. Each of these 
may be found by direct measurement, or from the known distance and the 
elevation, or depression, from the horizontal plane of the observer. If the 
two objects are on the same level as the observer, we have simply as in (234) 



Example. 

1865, May 16, 5f A. M. in lat 38° 15' 1ST., long. 76° 16' W.; 
the angular distance of the sun's centre from the top of a 
light-house measured by a sextant (©to the right of L. H.), 
75° 16' 25", index cor. —1' 15"; altitude of O above the 
sea-horizon observed at the same time, 10° 18' 20", index 
cor. +2' 10"; observed altitude of the top of light-house 
above the water-line, distant 7300 feet, 1° 15' 20", index cor., 
+ 2' 10"; height of eye, 20 feet; required the true bearing 
of the light-house. 



From the data we find 

O'sap. alt. H'.= 10° 31' 57"; ap. alt. of L. H. fc'= 1° 1' 34" 
O'strue" E = 10 27 7; ang. disk D = 75 15 10 

©'s dec +19 9 30. 





AZIMUTH OF A TERRESTRIAL OBJEC3 


283 




Computation 


(100) and (230). 




H = 


10° 27' 7" 1. sec 0.00726 


H' - 10° 31' 57" 


1. sec 0.00738 


L = 


38 15 l.sec 0.10496 


V = 1 7 34 


1. sec 0.00008 


P = 


70 50 30 


d = 9 24 23 




2s = 


119 32 37 


D =75 15 10 




s = 


59 46 18 1. cos 9.70196 


i (£+d) = 42 19 46 


L sin 9.82827 


p — s = 


11 4 12 1. cos 9.99184 


£(£— e*) = 32 55 24 


1. sin 9.73521 




19.80602 




19.57094 


iZ = 


36° 53'.0 1. cos 9.90301 


if = 37° 36'.2 
f= / 75 12.4 


1. sin 9.78547 




O 's azimuth 


Z=N. 73 46 .0E. 






True bearing of L. House 


(Z-J) = N. 1 26 .4 "W 





4 



